distr – Object Oriented Implementation of Distributions

Description

distr provides a conceptual treatment of distributions by means of S4 classes. A mother class Distribution is introduced with slots for a parameter and —most important— for the four constitutive methods r, d, p, and q for simulation respectively for evaluation of density / c.d.f.\ and quantile function of the corresponding distribution. Most distributions of package stats (like normal, Poisson, etc.) are implemented as subclasses of either AbscontDistribution or DiscreteDistribution, which themselves are again subclasses of Distribution. Up to arguments referring to a parameter of the distribution (like mean for the normal distribution), these function slots have the same arguments as those of package stats, i.e.; for a distribution object X we may call these functions as

• r(X)(n)

• d(X)(x, log = FALSE)

• p(X)(q, lower.tail = TRUE, log.p = FALSE)

• q(X)(p, lower.tail = TRUE, log.p = FALSE)

For the arguments of these function slots see e.g. rnorm. Note that, as usual, slots d, p, and q are vectorized in their first argument, but are not on the subsequent ones. In the environments of RStudio, see https://posit.co and Jupyter IRKernel, see https://github.com/IRkernel/IRkernel, calls to q are caught away from standard R evaluation and are treated in a non-standard way. This non-standard evaluation in particular throws errors at calls to our accessor methods q to slot q of the respective distribution object. To amend this, we provide function q.l as alias to our accessors q, so that our packages also become available in these environments. Arithmetics and unary mathematical transformations for distributions are available: For Distribution objects X and Y expressions like 3*X+sin(exp(-Y/4+3)) have their natural interpretation as corresponding image distributions.

Details

Classes

Distribution classes have a slot param the class of which is is specialized for the particualar distributions. The parameter classes for the particular distributions have slots with names according to the corresponding [rdpq]<name> functions of package base. From version 1.9 on, AbscontDistribution and descendants have a slot gaps for gaps in the support. DiscreteDistribution and descendants have an additional slot support, which is again specialized to be a lattice for class LatticeDistribution.
For saved objects from earlier versions, we provide the methods isOldVersion, and conv2NewVersion to check whether the object was generated by an older version of this package and to convert such an object to the new format, respectively. This applies to objects of subclasses of AbscontDistribution lacking a gap-slot as well as to to objects of subclasses of LatticeDistribution lacking a lattice-slot.
To enhance accuracy, from version 1.9 on, we also provide subclasses AffLinAbscontDistribution, AffLinDiscreteDistribution, and AffLinLatticeDistribution, as well as the class union AffLinDistribution, so that in particular functionals like E from package distrEx can recur to exact formula more frequently: These classes have additional slots a, b, and X0 to reflect the fact, that a distribution object of theses classes has the same distribution as a*X0+b.
For all particular distributions, as well as for classes AbscontDistribution, DiscreteDistribution, LatticeDistribution, UnivarDistrList and DistrList generating functions are provided, e.g. X <- Norm(mean = 3, sd = 2). The same goes for the space classes. All slots should be inspected / modified by means of corresponding accessor- /replacement functions; e.g. mean(X) <- 3 Again to enhance accuracy, from version 2.0 on, we also provide subclasses UnivarMixingDistribution to support mixing distributions, UnivarLebDecDistribution, to support Lebesgue decomposed distributions (with a discrete and an a.c. part) as well as AffLinUnivarLebDecDistribution, for corresponding affine linear transformations. Class UnivarLebDecDistribution is closed under arithmetical operations + /, *, ^ for pairs of independent variables + +, - for pairs of independent variables + affine linear transformations + truncation, huberization, min/max which are all now available analytically.
(see Parameter classes).

[*]: there is a generating function with the same name
##########################
Distribution classes
##########################
slots: [<name>(<class>)]
img(rSpace), param(OptionalParameter),
r(function), d(OptionalFunction), p(OptionalFunction), q(OptionalFunction),
.withSim(logical), .withArith(logical), .logExact(logical), .lowerExact(logical),
Symmetry(DistributionSymmetry)
"Distribution"
|>"UnivariateDistribution"
|>|>"UnivarMixingDistribution"            [*]
|>|>|>"UnivarLebDecDistribution"          [*]
|>|>|>|>"AffLinUnivarLebDecDistribution"
|>|>|>"CompoundDistribution"              [*]
|>|>"AbscontDistribution"                 [*]
|>|>|>"AffLinAbscontDistribution"
|>|>|>"Arcsine"                           [*]
|>|>|>"Beta"                              [*]
|>|>|>"Cauchy"                            [*]
|>|>|>"ExpOrGammaOrChisq" (VIRTUAL)
|>|>|>|>"Exp"                             [*]
|>|>|>|>"Gammad"                          [*]
|>|>|>|>"Chisq"                           [*]
|>|>|>"Fd"                                [*]
|>|>|>"Lnorm"                             [*]
|>|>|>"Logis"                             [*]
|>|>|>"Norm"                              [*]
|>|>|>"Td"                                [*]
|>|>|>"Unif"                              [*]
|>|>|>"Weibull"                           [*]
|>|>|"DiscreteDistribution"               [*]
|>|>|>"AffLinDiscreteDistribution"
|>|>|>"LatticeDistribution"               [*]
|>|>|>|>"AffLinLatticeDistribution"
|>|>|>|>"Binom"                           [*]
|>|>|>|>"Dirac"                           [*]
|>|>|>|>"Hyper"                           [*]
|>|>|>|>"NBinom"                          [*]
|>|>|>|>|>"Geom"                          [*]
|>|>|>|>"Pois"                            [*]
"AffLinDistribution" = union ( "AffLinAbscontDistribution",
                               "AffLinDiscreteDistribution",
                               "AffLinUnivarLebDecDistribution" )
"DistrList"
|>"UnivarDistrList"                       [*]
"AcDcLc" = union ( "AbscontDistribution",
                   "DiscreteDistribution",
                   "UnivarLebDecDistribution" )
##########################
Parameter classes
##########################
"OptionalParameter"
|>"Parameter"
|>|>"BetaParameter"
|>|>"BinomParameter"
|>|>"CauchyParameter"
|>|>"ChisqParameter"
|>|>"DiracParameter"
|>|>"ExpParameter"
|>|>"FParameter"
|>|>"GammaParameter"
|>|>"GeomParameter"
|>|>"HyperParameter"
|>|>"LnormParameter"
|>|>"LogisParameter"
|>|>"NbinomParameter"
|>|>"NormParameter"
|>|>"UniNormParameter"
|>|>|>"PoisParameter"
|>|>"TParameter"
|>|>"UnifParameter"
|>|>"WeibullParameter"
##########################
Space classes
##########################
"rSpace"
|>"EuclideanSpace"
|>|>"Reals"
|>"Lattice"
|>"Naturals"
##########################
Symmetry classes
##########################
slots:
type(character), SymmCenter(ANY)
"Symmetry"
|>"NoSymmetry"          [*]
|>"EllipticalSymmetry"  [*]
|>|>"SphericalSymmetry" [*]
|>"DistributionSymmetry"
|>"FunctionSymmetry"
|>|>"NonSymmetric"      [*]
|>|>"EvenSymmetric"     [*]
|>|>"OddSymmetric"      [*]
list thereof
"DistrSymmList"         [*]
"FunSymmList"           [*]
##########################
Matrix classes
##########################
slots:
none
"PosSemDefSymmMatrix" [*] is subclass of class "matrix" of package "base".
|>"PosDefSymmMatrix"  [*]
##########################
Class unions
##########################
"OptionalNumeric" = union("numeric", "NULL")
"OptionalMatrix" = union("matrix","NULL")

Methods

The group Math of unary (see Math) as well as convolution are made available for distributions, see operators-methods ;in particular for convolution powers, we have method convpow. Besides, there are plot and print-methods for distributions. For the space classes, we have liesIn, for the DicreteDistribution class, we have liesInSupport, as well as a generating function. The "history" of distributions obtained by chaining operations may be shortened using simplifyr.

Functions

RtoDPQ                  Default procedure to fill slots d,p,q given r
                        for a.c. distributions
RtoDPQ.d                Default procedure to fill slots d,p,q given r
                        for discrete distributions
RtoDPQ.LC               Default procedure to fill slots d,p,q given r
                        for Lebesgue decomposed distributions
decomposePM             decomposes a distribution into positive and negative
                        part and, if discrete, into part '0'
simplifyD               tries to reduce/simplify mixing distribution using
                        that certain weights are 0
flat.LCD                makes a single UnivarLebDecDistribution out of
                        a list of UnivarLebDecDistribution with corresp. weights
flat.mix                makes a single UnivarLebDecDistribution out of
                        a list of a UnivarMixingDistribution
distroptions            Functions to change the global variables of the
                        package 'distr'
standardMethods         Utility to automatically generate accessor and
                        replacement functions

Extension Packages in distrXXX family

Please note that there are extension packages of this packages available on CRAN,

distrDoc

a documentation package providing joint documentation for all packages of the distrXXX family of packages in the form of vignette 'distr'; try require(distrDoc); vignette("distr").

distrEx

provides functionals (like E, sd, mad) operating on distributions, as well as distances between distributions and basic support for multivariate and conditional distributions.

distrSim

for the standardized treatment of simulations, also under contaminations.

distrTEst

with classes and methods for evaluations of statistical procedures on simulations generated by distrSim.

distrTeach

embodies illustrations for basic stats courses using our distribution classes.

distrMod

provides classes for parametric models and hence covers, in an object orientated way, estimation in statistical models.

distrEllipse

provides classes for elliptically symmetric distributions.

Package versions

Note: The first two numbers of package versions do not necessarily reflect package-individual development, but rather are chosen for the distrXXX family as a whole in order to ease updating "depends" information.

Acknowledgement

We thank Martin Maechler, Josef Leydold, John Chambers, Duncan Murdoch, Gregory Warnes, Paul Gilbert, Kurt Hornik, Uwe Ligges, Torsten Hothorn, and Seth Falcon for their help in preparing this package.

Start-up-Banner

You may suppress the start-up banner/message completely by setting options("StartupBanner"="off") somewhere before loading this package by library or require in your R-code / R-session. If option "StartupBanner" is not defined (default) or setting options("StartupBanner"=NULL) or options("StartupBanner"="complete") the complete start-up banner is displayed. For any other value of option "StartupBanner" (i.e., not in c(NULL,"off","complete")) only the version information is displayed. The same can be achieved by wrapping the library or require call into either suppressStartupMessages() or onlytypeStartupMessages(.,atypes="version"). As for general packageStartupMessage's, you may also suppress all the start-up banner by wrapping the library or require call into suppressPackageStartupMessages() from startupmsg-version 0.5 on.

Demos

Demos are available — see demo(package="distr")

Note

Arithmetics on distribution objects are understood as operations on corresponding (independent) r.v.'s and not on distribution functions or densities.
See also distrARITH().
Some functions of package stats have intentionally been masked, but completely retain their functionality — see distrMASK().
Accuracy of these arithmetics is controlled by global options which may be inspected / set by distroptions() and getdistrOption(), confer distroptions .

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Matthias Kohl Matthias.Kohl@stamats.de
Maintainer: Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

References

P. Ruckdeschel, M. Kohl, T. Stabla, F. Camphausen (2006): S4 Classes for Distributions, R News, 6(2), 2-6. https://CRAN.R-project.org/doc/Rnews/Rnews_2006-2.pdf P. Ruckdeschel and M. Kohl (2014): General purpose convolution algorithm for distributions in S4-Classes by means of FFT. J. Statist. Softw. 59(4): 1-25. a vignette for packages distr, distrSim, distrTEst, and distrEx is included into the mere documentation package distrDoc and may be called by require("distrDoc");vignette("distr") a homepage to this package is available under
https://distr.r-forge.r-project.org/

Examples

X <- Unif(2,3)
Y <- Pois(lambda = 3)
Z <- X+Y  # generates Law of corresponding independent variables
p(Z)(0.2)
r(Z)(1000)
plot(Z+sin(Norm()))

Generating function "AbscontDistribution"

Description

Generates an object of class "AbscontDistribution"

Usage

AbscontDistribution(r = NULL, d = NULL, p = NULL, q = NULL,
                   gaps = NULL, param = NULL, img = new("Reals"),
                   .withSim = FALSE, .withArith = FALSE,
                    .lowerExact = FALSE, .logExact = FALSE,
                   withgaps = getdistrOption("withgaps"),
                   low1 = NULL, up1 = NULL, low = -Inf, up =Inf,
                   withStand = FALSE,
                   ngrid = getdistrOption("DefaultNrGridPoints"),
                   ep = getdistrOption("TruncQuantile"),
                   e = getdistrOption("RtoDPQ.e"),
                   Symmetry = NoSymmetry())

Arguments

Details

Typical usages are

  AbscontDistribution(r)
  AbscontDistribution(r = NULL, d)
  AbscontDistribution(r = NULL, d = NULL, p)
  AbscontDistribution(r = NULL, d = NULL, p = NULL, d)
  AbscontDistribution(r, d, p, q)
  

Minimally, only one of the slots r, d, p or q needs to be given as argument. The other non-given slots are then reconstructed according to the following scheme:

For this purpose, one may alternatively give arguments low1 and up1 (NULL each by default, and determined through slot q, resp. p, resp. d, resp. r in this order according to availability), for the (finite) range of values in the support of this distribution, as well as the possibly infinite theoretical range given by arguments low and up with default values -Inf, Inf, respectively. Of course all other slots may be specified as arguments.

Value

Object of class "AbscontDistribution"

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

AbscontDistribution-class, DiscreteDistribution-class, RtoDPQ

Examples

plot(Norm())
plot(AbscontDistribution(r = rnorm))
plot(AbscontDistribution(d = dnorm))
plot(AbscontDistribution(p = pnorm))
plot(AbscontDistribution(q = qnorm))
plot(Ac <- AbscontDistribution(d = function(x, log = FALSE){
                                   d <- exp(-abs(x^3))
                                   ## unstandardized!!
                                   if(log) d <- log(d)
                                   return(d)}, 
                         withStand = TRUE))

Class "AbscontDistribution"

Description

The AbscontDistribution-class is the mother-class of the classes Beta, Cauchy, Chisq, Exp, F, Gammad, Lnorm, Logis, Norm, T, Unif and Weibull. Further absolutely continuous distributions can be defined either by declaration of own random number generator, density, cumulative distribution and quantile functions, or as result of a convolution of two absolutely continuous distributions or by application of a mathematical operator to an absolutely continuous distribution.

Objects from the Class

Objects can be created by calls of the form new("AbscontDistribution", r, d, p, q). More comfortably, you may use the generating function AbscontDistribution. The result of these calls is an absolutely continuous distribution.

Slots

img

Object of class "Reals": the space of the image of this distribution which has dimension 1 and the name "Real Space"

param

Object of class "Parameter": the parameter of this distribution, having only the slot name "Parameter of an absolutely continuous distribution"

r

Object of class "function": generates random numbers

d

Object of class "function": density function

p

Object of class "function": cumulative distribution function

q

Object of class "function": quantile function

gaps

[from version 1.9 on] Object of class "OptionalMatrix", i.e.; an object which may either be NULL ora matrix. This slot, if non-NULL, contains left and right endpoints of intervals where the density of the object is 0. This slot may be inspected by the accessor gaps() and modified by a corresponding replacement method. It may also be filled automatically by setgaps(). For saved objects from earlier versions, we provide functions isOldVersion and conv2NewVersion.

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "UnivariateDistribution", directly.
Class "Distribution", by class "UnivariateDistribution".

Methods

initialize

signature(.Object = "AbscontDistribution"): initialize method

Math

signature(x = "AbscontDistribution"): application of a mathematical function, e.g. sin or exp (does not work with log, sign!), to this absolutely continouos distribution

• abs: signature(x = "AbscontDistribution"): exact image distribution of abs(x).

• exp: signature(x = "AbscontDistribution"): exact image distribution of exp(x).

• sign: signature(x = "AbscontDistribution"): exact image distribution of sign(x).

• sqrt: signature(x = "AbscontDistribution"): exact image distribution of sqrt(x).

• log: signature(x = "AbscontDistribution"): (with optional further argument base, defaulting to exp(1)) exact image distribution of log(x).

• log10: signature(x = "AbscontDistribution"): exact image distribution of log10(x).

• gamma: signature(x = "AbscontDistribution"): exact image distribution of gamma(x).

• lgamma: signature(x = "AbscontDistribution"): exact image distribution of lgamma(x).

• digamma: signature(x = "AbscontDistribution"): exact image distribution of digamma(x).

• sqrt: signature(x = "AbscontDistribution"): exact image distribution of sqrt(x).

-

signature(e1 = "AbscontDistribution"): application of ‘-’ to this absolutely continuous distribution.

*

signature(e1 = "AbscontDistribution", e2 = "numeric"): multiplication of this absolutely continuous distribution by an object of class "numeric"

/

signature(e1 = "AbscontDistribution", e2 = "numeric"): division of this absolutely continuous distribution by an object of class "numeric"

+

signature(e1 = "AbscontDistribution", e2 = "numeric"): addition of this absolutely continuous distribution to an object of class "numeric".

-

signature(e1 = "AbscontDistribution", e2 = "numeric"): subtraction of an object of class "numeric" from this absolutely continuous distribution.

*

signature(e1 = "numeric", e2 = "AbscontDistribution"): multiplication of this absolutely continuous distribution by an object of class "numeric".

+

signature(e1 = "numeric", e2 = "AbscontDistribution"): addition of this absolutely continuous distribution to an object of class "numeric".

-

signature(e1 = "numeric", e2 = "AbscontDistribution"): subtraction of this absolutely continuous distribution from an object of class "numeric".

+

signature(e1 = "AbscontDistribution", e2 = "AbscontDistribution"): Convolution of two absolutely continuous distributions. The slots p, d and q are approximated by grids.

-

signature(e1 = "AbscontDistribution", e2 = "AbscontDistribution"): Convolution of two absolutely continuous distributions. The slots p, d and q are approximated by grids.

plot

signature(object = "AbscontDistribution"): plots density, cumulative distribution and quantile function.

Internal subclass "AffLinAbscontDistribution"

To enhance accuracy of several functionals on distributions, mainly from package distrEx, from version 1.9 of this package on, there is an internally used (but exported) subclass "AffLinAbscontDistribution" which has extra slots a, b (both of class "numeric"), and X0 (of class "AbscontDistribution"), to capture the fact that the object has the same distribution as a * X0 + b. This is the class of the return value of methods

-

signature(e1 = "AbscontDistribution")

*

signature(e1 = "AbscontDistribution", e2 = "numeric")

/

signature(e1 = "AbscontDistribution", e2 = "numeric")

+

signature(e1 = "AbscontDistribution", e2 = "numeric")

-

signature(e1 = "AbscontDistribution", e2 = "numeric")

*

signature(e1 = "numeric", e2 = "AbscontDistribution")

+

signature(e1 = "numeric", e2 = "AbscontDistribution")

-

signature(e1 = "numeric", e2 = "AbscontDistribution")

-

signature(e1 = "AffLinAbscontDistribution")

*

signature(e1 = "AffLinAbscontDistribution", e2 = "numeric")

/

signature(e1 = "AffLinAbscontDistribution", e2 = "numeric")

+

signature(e1 = "AffLinAbscontDistribution", e2 = "numeric")

-

signature(e1 = "AffLinAbscontDistribution", e2 = "numeric")

*

signature(e1 = "numeric", e2 = "AffLinAbscontDistribution")

+

signature(e1 = "numeric", e2 = "AffLinAbscontDistribution")

-

signature(e1 = "numeric", e2 = "AffLinAbscontDistribution")

There also is a class union of "AffLinAbscontDistribution", "AffLinDiscreteDistribution", "AffLinUnivarLebDecDistribution" and called "AffLinDistribution" which is used for functionals.

Internal virtual superclass "AcDcLcDistribution"

As many operations should be valid no matter whether the operands are of class "AbscontDistribution", "DiscreteDistribution", or "UnivarLebDecDistribution", there is a class union of these classes called "AcDcLcDistribution"; in partiucalar methods for "*", "/", "^" (see operators-methods) and methods Minimum, Maximum, Truncate, and Huberize, and convpow are defined for this class union.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

AbscontDistribution Parameter-class UnivariateDistribution-class Beta-class Cauchy-class Chisq-class Exp-class Fd-class Gammad-class Lnorm-class Logis-class Norm-class Td-class Unif-class Weibull-class DiscreteDistribution-class Reals-class RtoDPQ

Examples

N <-  Norm() # N is a normal distribution with mean=0 and sd=1.
E <-  Exp() # E is an exponential distribution with rate=1.
A1 <-  E+1 # a new absolutely continuous distributions with exact slots d, p, q
A2 <-  A1*3 # a new absolutely continuous distributions with exact slots d, p, q
A3 <- N*0.9 + E*0.1 # a new absolutely continuous distribution with approximated slots d, p, q
r(A3)(1) # one random number generated from this distribution, e.g. -0.7150937
d(A3)(0) # The (approximated) density for x=0 is 0.43799.
p(A3)(0) # The (approximated) probability that x <= 0 is 0.45620.
q(A3)(.1) # The (approximated) 10 percent quantile is -1.06015.
## in RStudio or Jupytier IRKernel, use q.l(.)(.) instead of q(.)(.)

Class "Arcsine"

Description

The Arcsine distribution has density

f(x)=\frac{1}{\pi \sqrt{1-x^2}% }

for -1 < x < 1.

Objects from the Class

Objects can be created by calls of the form Arcsine(). This object is an Arcsine distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

r

Object of class "function": generates random numbers (calls function rArcsine)

d

Object of class "function": density function (calls function dArcsine)

p

Object of class "function": cumulative function (calls function pArcsine)

q

Object of class "function": inverse of the cumulative function (calls function qArcsine)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Methods

initialize

signature(.Object = "Arcsine"): initialize method

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

AbscontDistribution-class Reals-class

Examples

A <- Arcsine()
# A is a Arcsine distribution with shape1 = 1 and shape2 = 1.
r(A)(3) # three random number generated from this distribution, e.g. 0.6979795
d(A)(c(-2,-1,-0.2,0,0.2,1,2)) # Density at x=c(-1,-0.2,0,0.2,1).
p(A)(c(-2,-1,-0.2,0,0.2,1,2)) # cdf at q=c(-1,-0.2,0,0.2,1).
q(A)(c(0,0.2,1,2)) # quantile function at at x=c(0,0.2,1).
## in RStudio or Jupyter IRKernel, use q.l(A)(c(0,0.2,1,2)) instead

Class "Beta"

Description

The Beta distribution with parameters shape1 = a and shape2 = b has density

f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}{x}^{a-1} {(1-x)}^{b-1}%

for a > 0, b > 0 and 0 \le x \le 1 where the boundary values at x=0 or x=1 are defined as by continuity (as limits).

Ad hoc methods

For R Version <2.3.0 ad hoc methods are provided for slots q, r if ncp!=0; for R Version >=2.3.0 the methods from package stats are used.

Objects from the Class

Objects can be created by calls of the form Beta(shape1, shape2). This object is a beta distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "BetaParameter": the parameter of this distribution (shape1 and shape2), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rbeta)

d

Object of class "function": density function (calls function dbeta)

p

Object of class "function": cumulative function (calls function pbeta)

q

Object of class "function": inverse of the cumulative function (calls function qbeta)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Methods

initialize

signature(.Object = "Beta"): initialize method

shape1

signature(object = "Beta"): returns the slot shape1 of the parameter of the distribution

shape1<-

signature(object = "Beta"): modifies the slot shape1 of the parameter of the distribution

shape2

signature(object = "Beta"): returns the slot shape2 of the parameter of the distribution

shape2<-

signature(object = "Beta"): modifies the slot shape2 of the parameter of the distribution

-

signature(e1 = "numeric", e2 = "Beta") if ncp(e2)==0 and e1 == 1, an exact (central) Beta(shape1 = shape2(e2), shape2 = shape1(e2)) is returned, else the default method is used; exact

Note

The non-central Beta distribution is defined (Johnson et al, 1995, pp. 502) as the distribution of X/(X+Y) where X \sim \chi^2_{2a}(\lambda) and Y \sim \chi^2_{2b}. C.f. rbeta

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

BetaParameter-class AbscontDistribution-class Reals-class rbeta

Examples

B <- Beta(shape1 = 1, shape2 = 1)
# B is a beta distribution with shape1 = 1 and shape2 = 1.
r(B)(1) # one random number generated from this distribution, e.g. 0.6979795
d(B)(1) # Density of this distribution is 1 for x=1.
p(B)(1) # Probability that x < 1 is 1.
q(B)(.1) # Probability that x < 0.1 is 0.1.
shape1(B) # shape1 of this distribution is 1.
shape1(B) <- 2 # shape1 of this distribution is now 2.
Bn <- Beta(shape1 = 1, shape2 = 3, ncp = 5) 
# Bn is a beta distribution with shape1 = 1 and shape2 = 3 and ncp = 5.
B0 <- Bn; ncp(B0) <- 0; 
# B0 is just the same beta distribution as Bn but with ncp = 0
q(B0)(0.1) ## 
q(Bn)(0.1) ## => from R 2.3.0 on ncp no longer ignored...
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)

Class "BetaParameter"

Description

The parameter of a beta distribution, used by Beta-class

Objects from the Class

Objects can be created by calls of the form new("BetaParameter", shape1, shape2, ncp). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Beta is instantiated.

Slots

shape1

Object of class "numeric": the shape1 of a beta distribution

shape2

Object of class "numeric": the shape2 of a beta distribution

ncp

Object of class "numeric": the noncentrality parameter of a beta distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "BetaParameter"): initialize method

shape1

signature(object = "BetaParameter"): returns the slot shape1 of the parameter of the distribution

shape1<-

signature(object = "BetaParameter"): modifies the slot shape1 of the parameter of the distribution

shape2

signature(object = "BetaParameter"): returns the slot shape2 of the parameter of the distribution

shape2<-

signature(object = "BetaParameter"): modifies the slot shape2 of the parameter of the distribution

ncp

signature(object = "BetaParameter"): returns the slot ncp of the parameter of the distribution

ncp<-

signature(object = "BetaParameter"): modifies the slot ncp of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Beta-class Parameter-class

Examples

W <- new("BetaParameter", shape1 = 1, shape2 = 1, ncp = 0)
shape2(W) # shape2 of this distribution is 1.
shape2(W) <- 2 # shape2 of this distribution is now 2.

Class "Binom"

Description

The binomial distribution with size = n, by default =1, and prob = p, by default =0.5, has density

p(x) = {n \choose x} {p}^{x} {(1-p)}^{n-x}

for x = 0, \ldots, n.

C.f.rbinom

Objects from the Class

Objects can be created by calls of the form Binom(prob, size). This object is a binomial distribution.

Slots

img

Object of class "Naturals": The space of the image of this distribution has got dimension 1 and the name "Natural Space".

param

Object of class "BinomParameter": the parameter of this distribution (prob, size), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rbinom)

d

Object of class "function": density function (calls function dbinom)

p

Object of class "function": cumulative function (calls function pbinom)

q

Object of class "function": inverse of the cumulative function (calls function qbinom). The quantile is defined as the smallest value x such that F(x) >= p, where F is the cumulative function.

support

Object of class "numeric": a (sorted) vector containing the support of the discrete density function

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "DiscreteDistribution", directly.
Class "UnivariateDistribution", by class "DiscreteDistribution".
Class "Distribution", by class "DiscreteDistribution".

Methods

+

signature(e1 = "Binom", e2 = "Binom"): For two binomial distributions with equal probabilities the exact convolution formula is implemented thereby improving the general numerical accuracy.

initialize

signature(.Object = "Binom"): initialize method

prob

signature(object = "Binom"): returns the slot prob of the parameter of the distribution

prob<-

signature(object = "Binom"): modifies the slot prob of the parameter of the distribution

size

signature(object = "Binom"): returns the slot size of the parameter of the distribution

size<-

signature(object = "Binom"): modifies the slot size of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

BinomParameter-class DiscreteDistribution-class Naturals-class rbinom

Examples

B <- Binom(prob=0.5,size=1) # B is a binomial distribution with prob=0.5 and size=1.
r(B)(1) # # one random number generated from this distribution, e.g. 1
d(B)(1) # Density of this distribution is  0.5 for x=1.
p(B)(0.4) # Probability that x<0.4 is 0.5.
q(B)(.1) # x=0 is the smallest value x such that p(B)(x)>=0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
size(B) # size of this distribution is 1.
size(B) <- 2 # size of this distribution is now 2.
C <- Binom(prob = 0.5, size = 1) # C is a binomial distribution with prob=0.5 and size=1.
D <- Binom(prob = 0.6, size = 1) # D is a binomial distribution with prob=0.6 and size=1.
E <- B + C # E is a binomial distribution with prob=0.5 and size=3.
F <- B + D # F is an object of class LatticeDistribution.
G <- B + as(D,"DiscreteDistribution") ## DiscreteDistribution

Class "BinomParameter"

Description

The parameter of a binomial distribution, used by Binom-class

Objects from the Class

Objects can be created by calls of the form new("BinomParameter", prob, size). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Binom is instantiated.

Slots

prob

Object of class "numeric": the probability of a binomial distribution

size

Object of class "numeric": the size of a binomial distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "BinomParameter"): initialize method

prob

signature(object = "BinomParameter"): returns the slot prob of the parameter of the distribution

prob<-

signature(object = "BinomParameter"): modifies the slot prob of the parameter of the distribution

size

signature(object = "BinomParameter"): returns the slot size of the parameter of the distribution

size<-

signature(object = "BinomParameter"): modifies the slot size of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Binom-class Parameter-class

Examples

W <- new("BinomParameter",prob=0.5,size=1)
size(W) # size of this distribution is 1.
size(W) <- 2 # size of this distribution is now 2.

Class "Cauchy"

Description

The Cauchy distribution with location l, by default =0, and scale s , by default =1,has density

f(x) = \frac{1}{\pi s} \left( 1 + \left(\frac{x - l}{s}\right)^2 \right)^{-1}%

for all x. C.f. rcauchy

Objects from the Class

Objects can be created by calls of the form Cauchy(location, scale). This object is a Cauchy distribution.

Slots

img

Object of class "Reals": The domain of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "CauchyParameter": the parameter of this distribution (location and scale), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rcauchy)

d

Object of class "function": density function (calls function dcauchy)

p

Object of class "function": cumulative function (calls function pcauchy)

q

Object of class "function": inverse of the cumulative function (calls function qcauchy)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Is-Relations

By means of setIs, R “knows” that a distribution object obj of class "Cauchy" with location 0 and scale 1 also is a T distribution with parameters df = 1, ncp = 0.

Methods

initialize

signature(.Object = "Cauchy"): initialize method

location

signature(object = "Cauchy"): returns the slot location of the parameter of the distribution

location<-

signature(object = "Cauchy"): modifies the slot location of the parameter of the distribution

scale

signature(object = "Cauchy"): returns the slot scale of the parameter of the distribution

scale<-

signature(object = "Cauchy"): modifies the slot scale of the parameter of the distribution

+

signature(e1 = "Cauchy", e2 = "Cauchy"): For the Cauchy distribution the exact convolution formula is implemented thereby improving the general numerical approximation.

*

signature(e1 = "Cauchy", e2 = "numeric")

+

signature(e1 = "Cauchy", e2 = "numeric"): For the Cauchy location scale family we use its closedness under affine linear transformations.

further arithmetic methods see operators-methods

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

CauchyParameter-class AbscontDistribution-class Reals-class rcauchy

Examples

C <- Cauchy(location = 1, scale = 1) # C is a Cauchy distribution with location=1 and scale=1.
r(C)(1) # one random number generated from this distribution, e.g. 4.104603
d(C)(1) # Density of this distribution is 0.3183099 for x=1.
p(C)(1) # Probability that x<1 is 0.5.
q(C)(.1) # Probability that x<-2.077684 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
location(C) # location of this distribution is 1.
location(C) <- 2 # location of this distribution is now 2.
is(C,"Td") # no
C0 <- Cauchy() # standard, i.e. location = 0, scale = 1
is(C0,"Td") # yes
as(C0,"Td") 

Class "CauchyParameter"

Description

The parameter of a Cauchy distribution, used by Cauchy-class

Objects from the Class

Objects can be created by calls of the form new("CauchyParameter", location, scale). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Cauchy is instantiated.

Slots

location:

Object of class "numeric": the location of a Cauchy distribution

scale

Object of class "numeric": the scale of a Cauchy distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "CauchyParameter"): initialize method

scale

signature(object = "CauchyParameter"): returns the slot scale of the parameter of the distribution

scale<-

signature(object = "CauchyParameter"): modifies the slot scale of the parameter of the distribution

location

signature(object = "CauchyParameter"): returns the slot location of the parameter of the distribution

location<-

signature(object = "CauchyParameter"): modifies the slot location of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Cauchy-class Parameter-class

Examples

W <- new("CauchyParameter",location=1,scale=1)
location(W) # location of this distribution is 1.
location(W) <- 2 # location of this distribution is now 2.

Class "Chisq"

Description

The chi-squared distribution with df= n degrees of freedom has density

f_n(x) = \frac{1}{{2}^{n/2} \Gamma (n/2)} {x}^{n/2-1} {e}^{-x/2}

for x > 0. The mean and variance are n and 2n.

The non-central chi-squared distribution with df= n degrees of freedom and non-centrality parameter ncp = \lambda has density

f(x) = e^{-\lambda / 2} \sum_{r=0}^\infty \frac{(\lambda/2)^r}{r!}\, f_{n + 2r}(x)

for x \ge 0. For integer n, this is the distribution of the sum of squares of n normals each with variance one, \lambda being the sum of squares of the normal means.

C.f. rchisq

Objects from the Class

Objects can be created by calls of the form Chisq(df, ncp). This object is a chi-squared distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "ChisqParameter": the parameter of this distribution (df and ncp), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rchisq)

d

Object of class "function": density function (calls function dchisq)

p

Object of class "function": cumulative function (calls function pchisq)

q

Object of class "function": inverse of the cumulative function (calls function qchisq)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "ExpOrGammaOrChisq", directly.
Class "AbscontDistribution", by class "ExpOrGammaOrChisq".
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "UnivariateDistribution".

Is-Relations

By means of setIs, R “knows” that a distribution object obj of class "Chisq" with non-centrality 0 also is a Gamma distribution with parameters shape = df(obj)/2, scale = 2.

Methods

initialize

signature(.Object = "Chisq"): initialize method

df

signature(object = "Chisq"): returns the slot df of the parameter of the distribution

df<-

signature(object = "Chisq"): modifies the slot df of the parameter of the distribution

ncp

signature(object = "Chisq"): returns the slot ncp of the parameter of the distribution

ncp<-

signature(object = "Chisq"): modifies the slot ncp of the parameter of the distribution

+

signature(e1 = "Chisq", e2 = "Chisq"): For the chi-squared distribution we use its closedness under convolutions.

Note

Warning: The code for pchisq and qchisq is unreliable for values of ncp above approximately 290.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

ChisqParameter-class AbscontDistribution-class Reals-class rchisq

Examples

C <- Chisq(df = 1, ncp = 1) # C is a chi-squared distribution with df=1 and ncp=1.
r(C)(1) # one random number generated from this distribution, e.g. 0.2557184
d(C)(1) # Density of this distribution is 0.2264666 for x = 1.
p(C)(1) # Probability that x < 1 is 0.4772499.
q(C)(.1) # Probability that x < 0.04270125 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
df(C) # df of this distribution is 1.
df(C) <- 2 # df of this distribution is now 2.
is(C, "Gammad") # no
C0 <- Chisq() # default: Chisq(df=1,ncp=0)
is(C0, "Gammad") # yes
as(C0,"Gammad")

Class "ChisqParameter"

Description

The parameter of a chi-squared distribution, used by Chisq-class

Objects from the Class

Objects can be created by calls of the form new("ChisqParameter", ncp, df). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Chisq is instantiated.

Slots

ncp

Object of class "numeric": the ncp of a chi-squared distribution

df

Object of class "numeric": the df of a chi-squared distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "ChisqParameter"): initialize method

df

signature(object = "ChisqParameter"): returns the slot df of the parameter of the distribution

df<-

signature(object = "ChisqParameter"): modifies the slot df of the parameter of the distribution

ncp

signature(object = "ChisqParameter"): returns the slot ncp of the parameter of the distribution

ncp<-

signature(object = "ChisqParameter"): modifies the slot ncp of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Chisq-class Parameter-class

Examples

W <- new("ChisqParameter",df=1,ncp=1)
ncp(W) # ncp of this distribution is 1.
ncp(W) <- 2 # ncp of this distribution is now 2.

Generating function for Class "CompoundDistribution"

Description

Generates an object of class "CompoundDistribution".

Usage

CompoundDistribution(NumbOfSummandsDistr, SummandsDistr, .withSim = FALSE,
                                 withSimplify = FALSE)
                                

Arguments

Value

Object of class "CompoundDistribution", or if argument withSimplify is TRUE the result of simplifyD applied to the compound distribution, i.e. an object of class "UnivarLebDecDistribution", or if degenerate, of class "AbscontDistribution" or "DiscreteDistribution".

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

CompoundDistribution-class, simplifyD

Examples

CP0 <- CompoundDistribution(Pois(), Norm())
CP0
CP1 <- CompoundDistribution(DiscreteDistribution(supp = c(1,5,9,11),
                            prob = dbinom(0:3, size = 3,prob = 0.3)),Norm())
CP1
UL <- UnivarDistrList(Norm(), Binom(10,0.3), Chisq(df=4), Norm(),
                      Binom(10,0.3), Chisq(df=4), Norm(), Binom(10,0.3),
                      Chisq(df=4), Td(5), Td(10))
CP2 <- CompoundDistribution(DiscreteDistribution(supp = c(1,5,9,11),
                      prob = dbinom(0:3, size = 3, prob = 0.3)),UL)
plot(CP2)

Class "CompoundDistribution"

Description

CompoundDistribution-class is a class to formalize compound distributions; it is a subclass to class UnivarMixingDistribution.

Objects from the Class

Objects can be created by calls of the form new("CompoundDistribution", ...). More frequently they are created via the generating function CompoundDistribution.

Slots

NumbOfSummandsDistr

Object of class "DiscreteDistribution", the frequency distribution.

SummandsDistr

Object of class "UnivDistrListOrDistribution", that is, either of class "UnivarDistrList" (non i.i.d. case) or of class "UnivariateDistribution" (i.i.d. case); the summand distribution(s).

mixCoeff

Object of class "numeric": a vector of probabilities for the mixing components.

mixDistr

Object of class "UnivarDistrList": a list of univariate distributions containing the mixing components; must be of same length as mixCoeff.

img

Object of class "Reals": the space of the image of this distribution which has dimension 1 and the name "Real Space"

param

Object of class "Parameter": the parameter of this distribution, having only the slot name "Parameter of a discrete distribution"

r

Object of class "function": generates random numbers

d

fixed to NULL

p

Object of class "function": cumulative distribution function

q

Object of class "function": quantile function

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "UnivarMixingDistribution" class "UnivarDistribution" by class "UnivarMixingDistribution", class "Distribution" by class "UnivariateDistribution".

Methods

show

signature(object = "CompoundDistribution") prints the object

SummandsDistr

signature(object = "CompoundDistribution") returns the corresponding slot

NumbOfSummandsDistr

signature(object = "CompoundDistribution") returns the corresponding slot

setAs relations

There is a coerce method to coerce objects of class "CompoundDistribution" to class UnivarLebDecDistribution; this is done by a simple call to simplifyD.

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

Parameter-class, UnivariateDistribution-class, LatticeDistribution-class, AbscontDistribution-class, simplifyD, flat.mix

Examples

CP <- CompoundDistribution(Pois(),Norm())
CP
p(CP)(0.3)          
plot(CP)

Class "DExp"

Description

The double exponential or Laplace distribution with rate \lambda has density

f(x) = \frac{1}{2}\lambda {e}^{- \lambda |x|}

C.f. Exp-class, rexp

Objects from the Class

Objects can be created by calls of the form DExp(rate). This object is a double exponential (or Laplace) distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "ExpParameter": the parameter of this distribution (rate), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rexp)

d

Object of class "function": density function (calls function dexp)

p

Object of class "function": cumulative function (calls function pexp)

q

Object of class "function": inverse of the cumulative function (calls function qexp)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution". Class "Distribution", by class "AbscontDistribution".

Methods

initialize

signature(.Object = "DExp"): initialize method

rate

signature(object = "DExp"): returns the slot rate of the parameter of the distribution

rate<-

signature(object = "DExp"): modifies the slot rate of the parameter of the distribution

*

signature(e1 = "DExp", e2 = "numeric"): For the Laplace distribution we use its closedness under scaling transformations.

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

Exp-class ExpParameter-class AbscontDistribution-class Reals-class rexp

Examples

D <- DExp(rate = 1) # D is a Laplace distribution with rate = 1.
r(D)(1) # one random number generated from this distribution, e.g. 0.4190765
d(D)(1) # Density of this distribution is 0.1839397 for x = 1.
p(D)(1) # Probability that x < 1 is 0.8160603.
q(D)(.1) # Probability that x < -1.609438 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
rate(D) # rate of this distribution is 1.
rate(D) <- 2 # rate of this distribution is now 2.
3*D ###  still a DExp -distribution

Class "Dirac"

Description

The Dirac distribution with location l, by default =0, has density d(x) = 1 for x = l, 0 else.

Objects from the Class

Objects can be created by calls of the form Dirac(location). This object is a Dirac distribution.

Slots

img

Object of class "Naturals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "DiracParameter": the parameter of this distribution (location), declared at its instantiation

r

Object of class "function": generates random numbers

d

Object of class "function": density function

p

Object of class "function": cumulative function

q

Object of class "function": inverse of the cumulative function

support

Object of class "numeric": a (sorted) vector containing the support of the discrete density function

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "DiscreteDistribution", directly.
Class "UnivariateDistribution", by class "DiscreteDistribution".
Class "Distribution", by class "DiscreteDistribution".

Methods

-

signature(e1 = "Dirac", e2 = "Dirac")

+

signature(e1 = "Dirac", e2 = "Dirac")

*

signature(e1 = "Dirac", e2 = "Dirac")

/

signature(e1 = "Dirac", e2 = "Dirac"): For the Dirac distribution these operations are trivial.

initialize

signature(.Object = "Dirac"): initialize method

location

signature(object = "Dirac"): returns the slot location of the parameter of the distribution

location<-

signature(object = "Dirac"): modifies the slot location of the parameter of the distribution

log

signature(object = "Dirac"): returns an object of class "Dirac" distribution with log-transformed location parameter.

Math

signature(object = "Dirac"): given a "Math" group generic fun an object of class "Dirac" distribution with fun-transformed location parameter is returned.

further arithmetic methods see operators-methods

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

DiracParameter-class DiscreteDistribution-class Naturals-class

Examples

D <- Dirac(location = 0) # D is a Dirac distribution with location=0.
r(D)(1)
# r(D)(1) generates a pseudo-random-number according to a Dirac
# distribution with location = 0,
# which of course will take 0 as value almost surely.
d(D)(0) # Density of this distribution is 1 for x = 0.
p(D)(1) # Probability that x < 1 is 1.
q(D)(.1) # q(D)(x) is always 0 (= location).
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
location(D) # location of this distribution is 0.
location(D) <- 2 # location of this distribution is now 2.

Class "DiracParameter"

Description

The parameter of a Dirac distribution, used by Dirac-class

Objects from the Class

Objects can be created by calls of the form new("DiracParameter", location). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Dirac is instantiated.

Slots

location

Object of class "numeric": the location of a Dirac distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "DiracParameter"): initialize method

location

signature(object = "DiracParameter"): returns the slot location of the parameter of the distribution

location<-

signature(object = "DiracParameter"): modifies the slot location of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Dirac-class Parameter-class

Examples

W <- new("DiracParameter",location=1)
location(W) # location of this distribution is 1.
location(W) <- 2 # location of this distribution is now 2.

Generating function "DiscreteDistribution"

Description

Generates an object of class "DiscreteDistribution"

Usage

  DiscreteDistribution(supp, prob, .withArith=FALSE, .withSim=FALSE, 
                       .lowerExact = TRUE, .logExact = FALSE,
             .DistrCollapse = getdistrOption("DistrCollapse"),
             .DistrCollapse.Unique.Warn = 
                  getdistrOption("DistrCollapse.Unique.Warn"),
             .DistrResolution = getdistrOption("DistrResolution"),
             Symmetry = NoSymmetry())

Arguments

Details

If prob is missing, all elements in supp are equally weighted.

Typical usages are

    DiscreteDistribution(supp, prob)
    DiscreteDistribution(supp)
  

Value

Object of class "DiscreteDistribution"

Note

Working with a computer, we use a finite interval as support which carries at least mass 1-getdistrOption("TruncQuantile").

Also, we require that support points have distance at least .DistrResoltion, if this condition fails, upon a suggestion by Jacob van Etten, jacobvanetten@yahoo.com, we use the global option .DistrCollapse to decide whether we use collapsing or not. If we do so, we collapse support points if they are too close to each other, taking the (left most) median among them as new support point which accumulates all the mass of the collapsed points. With .DistrCollapse==FALSE, we at least collapse points according to the result of unique(), and if after this collapsing, the minimal distance is less than .DistrResoltion, we throw an error. By .DistrCollapse.Unique.Warn, we control, whether we throw a warning upon collapsing or not.

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

DiscreteDistribution-class AbscontDistribution-class RtoDPQ.d

Examples

# Dirac-measure at 0
D1 <- DiscreteDistribution(supp = 0)
D1
# simple discrete distribution
D2 <- DiscreteDistribution(supp = c(1:5), prob = c(0.1, 0.2, 0.3, 0.2, 0.2))
D2

plot(D2)

Class "DiscreteDistribution"

Description

The DiscreteDistribution-class is the mother-class of the class LatticeDistribution.

Objects from the Class

Objects can be created by calls to new("DiscreteDistribution", ...), but more easily is the use of the generating function "DiscreteDistribution". This generating function, from version 1.9 on, has been moved to this package from package distrEx.

Slots

img

Object of class "Reals": the space of the image of this distribution which has dimension 1 and the name "Real Space"

param

Object of class "Parameter": the parameter of this distribution, having only the slot name "Parameter of a discrete distribution"

r

Object of class "function": generates random numbers

d

Object of class "function": density/probability function

p

Object of class "function": cumulative distribution function

q

Object of class "function": quantile function

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

.finSupport

logical: used internally to check whether the true support is finite; in case img is one-dimensional, it is of length 2 (left and right end).

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "UnivariateDistribution", directly.
Class "Distribution", by class "UnivariateDistribution".

Methods

initialize

signature(.Object = "DiscreteDistribution"): initialize method

coerce

signature(from = "DiscreteDistribution", to = "LatticeDistribution"): coerce method to class "LatticeDistribution" (checks if support is a lattice)

Math

signature(x = "DiscreteDistribution"): application of a mathematical function, e.g. sin or tan to this discrete distribution

• abs: signature(x = "DiscreteDistribution"): exact image distribution of abs(x).

• exp: signature(x = "DiscreteDistribution"): exact image distribution of exp(x).

• sign: signature(x = "DiscreteDistribution"): exact image distribution of sign(x).

• sqrt: signature(x = "DiscreteDistribution"): exact image distribution of sqrt(x).

• log: signature(x = "DiscreteDistribution"): (with optional further argument base, defaulting to exp(1)) exact image distribution of log(x).

• log10: signature(x = "DiscreteDistribution"): exact image distribution of log10(x).

• gamma: signature(x = "DiscreteDistribution"): exact image distribution of gamma(x).

• lgamma: signature(x = "DiscreteDistribution"): exact image distribution of lgamma(x).

• digamma: signature(x = "DiscreteDistribution"): exact image distribution of digamma(x).

-

signature(e1 = "DiscreteDistribution"): application of ‘-’ to this discrete distribution

*

signature(e1 = "DiscreteDistribution", e2 = "numeric"): multiplication of this discrete distribution by an object of class ‘numeric’

/

signature(e1 = "DiscreteDistribution", e2 = "numeric"): division of this discrete distribution by an object of class ‘numeric’

+

signature(e1 = "DiscreteDistribution", e2 = "numeric"): addition of this discrete distribution to an object of class ‘numeric’

-

signature(e1 = "DiscreteDistribution", e2 = "numeric"): subtraction of an object of class ‘numeric’ from this discrete distribution

*

signature(e1 = "numeric", e2 = "DiscreteDistribution"): multiplication of this discrete distribution by an object of class ‘numeric’

+

signature(e1 = "numeric", e2 = "DiscreteDistribution"): addition of this discrete distribution to an object of class ‘numeric’

-

signature(e1 = "numeric", e2 = "DiscreteDistribution"): subtraction of this discrete distribution from an object of class ‘numeric’

+

signature(e1 = "DiscreteDistribution", e2 = "DiscreteDistribution"): Convolution of two discrete distributions. The slots p, d and q are approximated on a common grid.

-

signature(e1 = "DiscreteDistribution", e2 = "DiscreteDistribution"): Convolution of two discrete distributions. The slots p, d and q are approximated on a common grid.

support

signature(object = "DiscreteDistribution"): returns the support

p.l

signature(object = "DiscreteDistribution"): returns the left continuous cumulative distribution function, i.e.; p.l(t) = P(object < t)

q.r

signature(object = "DiscreteDistribution"): returns the right-continuous quantile function, i.e.; {\rm q.r}(s)=\sup\{t \,\big|\, P({\tt object}\ge t)\leq s\}

plot

signature(object = "DiscreteDistribution"): plots density, cumulative distribution and quantile function

Internal subclass "AffLinDiscreteDistribution"

To enhance accuracy of several functionals on distributions, mainly from package distrEx, from version 1.9 of this package on, there is an internally used (but exported) subclass "AffLinDiscreteDistribution" which has extra slots a, b (both of class "numeric"), and X0 (of class "DiscreteDistribution"), to capture the fact that the object has the same distribution as a * X0 + b. This is the class of the return value of methods

-

signature(e1 = "DiscreteDistribution")

*

signature(e1 = "DiscreteDistribution", e2 = "numeric")

/

signature(e1 = "DiscreteDistribution", e2 = "numeric")

+

signature(e1 = "DiscreteDistribution", e2 = "numeric")

-

signature(e1 = "DiscreteDistribution", e2 = "numeric")

*

signature(e1 = "numeric", e2 = "DiscreteDistribution")

+

signature(e1 = "numeric", e2 = "DiscreteDistribution")

-

signature(e1 = "numeric", e2 = "DiscreteDistribution")

-

signature(e1 = "AffLinDiscreteDistribution")

*

signature(e1 = "AffLinDiscreteDistribution", e2 = "numeric")

/

signature(e1 = "AffLinDiscreteDistribution", e2 = "numeric")

+

signature(e1 = "AffLinDiscreteDistribution", e2 = "numeric")

-

signature(e1 = "AffLinDiscreteDistribution", e2 = "numeric")

*

signature(e1 = "numeric", e2 = "AffLinDiscreteDistribution")

+

signature(e1 = "numeric", e2 = "AffLinDiscreteDistribution")

-

signature(e1 = "numeric", e2 = "AffLinDiscreteDistribution")

There also is a class union of "AffLinAbscontDistribution", "AffLinDiscreteDistribution", "AffLinUnivarLebDecDistribution" and called "AffLinDistribution" which is used for functionals.

Internal virtual superclass "AcDcLcDistribution"

As many operations should be valid no matter whether the operands are of class "AbscontDistribution", "DiscreteDistribution", or "UnivarLebDecDistribution", there is a class union of these classes called "AcDcLcDistribution"; in partiucalar methods for "*", "/", "^" (see operators-methods) and methods Minimum, Maximum, Truncate, and Huberize, and convpow are defined for this class union.

Note

Working with a computer, we use a finite interval as support which carries at least mass 1-getdistrOption("TruncQuantile").

Also, we require that support points have distance at least getdistrOption("DistrResoltion"), if this condition fails, upon a suggestion by Jacob van Etten, jacobvanetten@yahoo.com, we use the global option getdistrOption("DistrCollapse") to decide whether we use collapsing or not. If we do so, we collapse support points if they are too close to each other, taking the (left most) median among them as new support point which accumulates all the mass of the collapsed points. With getdistrOption("DistrCollapse")==FALSE, we at least collapse points according to the result of unique(), and if after this collapsing, the minimal distance is less than getdistrOption("DistrResoltion"), we throw an error. By getdistrOption("DistrCollapse.Unique.Warn"), we control, whether we throw a warning upon collapsing or not.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Parameter-class UnivariateDistribution-class LatticeDistribution-class AbscontDistribution-class Reals-class RtoDPQ.d

Examples

# Dirac-measure at 0
D1 <- DiscreteDistribution(supp = 0)
support(D1)

# simple discrete distribution
D2 <- DiscreteDistribution(supp = c(1:5), prob = c(0.1, 0.2, 0.3, 0.2, 0.2))
plot(D2)
(pp <- p(D2)(support(D2)))
p(D2)(support(D2)-1e-5)
p(D2)(support(D2)+1e-5)
p.l(D2)(support(D2))
p.l(D2)(support(D2)-1e-5)
p.l(D2)(support(D2)+1e-5)
q(D2)(pp)
q(D2)(pp-1e-5)
q(D2)(pp+1e-5)
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
q.r(D2)(pp)
q.r(D2)(pp-1e-5)
q.r(D2)(pp+1e-5)

Generating function for DistrList-class

Description

Generates an object of class "DistrList".

Usage

DistrList(..., Dlist)

Arguments

Value

Object of class "DistrList"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

DistrList-class, UnivarDistrList-class, UnivarDistrList

Examples

(DL <- DistrList(Norm(), Exp(), Pois()))
plot(DL)
as(Norm(), "DistrList")

## The function is currently defined as
function(...){ 
    new("DistrList", list(...)) 
}

List of distributions

Description

Create a list of distributions

Objects from the Class

Objects can be created by calls of the form new("DistrList", ...). More frequently they are created via the generating function DistrList.

Slots

.Data

Object of class "list". A list of distributions.

Extends

Class "list", from data part.
Class "vector", by class "list".

Methods

show

signature(object = "DistrList")

plot

signature(object = "DistrList")

coerce

signature(from = "Distribution", to = "DistrList"): create a "DistrList" object from a "Distribution" object

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

DistrList, Distribution-class

Examples

(DL <- new("DistrList", list(Norm(), Exp())))
plot(DL)
as(Norm(), "DistrList")

Generating function for DistrSymmList-class

Description

Generates an object of class "DistrSymmList".

Usage

DistrSymmList(...)

Arguments

Value

Object of class "DistrSymmList"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

DistrSymmList-class

Examples

DistrSymmList(NoSymmetry(), SphericalSymmetry(SymmCenter = 1), 
              EllipticalSymmetry(SymmCenter = 2))

## The function is currently defined as
function (...){
    new("DistrSymmList", list(...))
}

List of Symmetries for a List of Distributions

Description

Create a list of symmetries for a list of distributions

Objects from the Class

Objects can be created by calls of the form new("DistrSymmList", ...). More frequently they are created via the generating function DistrSymmList.

Slots

.Data

Object of class "list". A list of objects of class "DistributionSymmetry".

Extends

Class "list", from data part.
Class "vector", by class "list".

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

DistributionSymmetry-class

Examples

new("DistrSymmList", list(NoSymmetry(), SphericalSymmetry(SymmCenter = 1), 
                          EllipticalSymmetry(SymmCenter = 2)))

Class "Distribution"

Description

The Distribution-class is the mother-class of class UnivariateDistribution.

Objects from the Class

Objects can be created by calls of the form new("Distribution").

Slots

img

Object of class "rSpace": the space of the image

param

Object of class "OptionalParameter": the parameter

r

Object of class "function": generates random numbers

d

Object of class "OptionalFunction": density function

p

Object of class "OptionalFunction": cumulative distribution function

q

Object of class "OptionalFunction": quantile function

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Methods

img

signature(object = "Distribution"): returns the space of the image

param

signature(object = "Distribution"): returns the parameter

r

signature(object = "Distribution"): returns the random number generator

d

signature(object = "Distribution"): returns the density function

p

signature(object = "Distribution"): returns the cumulative distribution function

q

signature(object = "Distribution"): returns the quantile function

.logExact

signature(object = "Distribution"): returns slot .logExact if existing; else tries to convert the object to a newer version of its class by conv2NewVersion and returns the corresponding slot of the converted object.

.lowerExact

signature(object = "Distribution"): returns slot .lowerExact if existing; else tries to convert the object to a newer version of its class by conv2NewVersion and returns the corresponding slot of the converted object.

Symmetry:

returns slot Symmetry if existing; else tries to convert the object to a newer version of its class by conv2NewVersion and returns the corresponding slot of the converted object.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

UnivariateDistribution-class Parameter-class

Class of Symmetries for Distributions

Description

Class of symmetries for distributions.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

type

Object of class "character": discribes type of symmetry.

SymmCenter

Object of class "OptionalNumeric": center of symmetry.

Extends

Class "Symmetry", directly.

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

Symmetry-class, Distribution-class, OptionalNumeric-class

Generating function for EllipticalSymmetry-class

Description

Generates an object of class "EllipticalSymmetry".

Usage

EllipticalSymmetry(SymmCenter = 0)

Arguments

Value

Object of class "EllipticalSymmetry"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

EllipticalSymmetry-class, DistributionSymmetry-class

Examples

EllipticalSymmetry()

## The function is currently defined as
function(SymmCenter = 0){ 
    new("EllipticalSymmetry", SymmCenter = SymmCenter) 
}

Class for Elliptically Symmetric Distributions

Description

Class for elliptically symmetric distributions.

Objects from the Class

Objects can be created by calls of the form new("EllipticalSymmetry"). More frequently they are created via the generating function EllipticalSymmetry. Elliptical symmetry for instance leads to a simplification for the computation of optimally robust influence curves.

Slots

type

Object of class "character": contains “elliptical symmetric distribution”

SymmCenter

Object of class "numeric": center of symmetry

Extends

Class "DistributionSymmetry", directly.
Class "Symmetry", by class "DistributionSymmetry".

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

EllipticalSymmetry, DistributionSymmetry-class

Examples

new("EllipticalSymmetry")

Generating function "EmpiricalDistribution"

Description

Generates an object of class "DiscreteDistribution"

Usage

  EmpiricalDistribution(data, .withArith=FALSE, .withSim=FALSE, 
                        .lowerExact = TRUE, .logExact = FALSE,
                        .DistrCollapse = getdistrOption("DistrCollapse"),
                        .DistrCollapse.Unique.Warn = 
                             getdistrOption("DistrCollapse.Unique.Warn"),
                        .DistrResolution = getdistrOption("DistrResolution"),
                        Symmetry = NoSymmetry())

Arguments

Details

The function is a simple utility function providing a wrapper to the generating function DiscreteDistribution.

Typical usage is

    EmpiricalDistribution(data)
  

Value

Object of class "DiscreteDistribution"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

DiscreteDistribution DiscreteDistribution-class

Examples

x <- rnorm(20)
D1 <- EmpiricalDistribution(data = x)
D1

plot(D1)

Class "EuclideanSpace"

Description

The distribution-classes contain a slot where the sample space is stored. One typical sample space is the Euclidean Space in dimension k.

Usage

EuclideanSpace(dimension = 1)

Arguments

Objects from the Class

Objects could theoretically be created by calls of the form new("EuclideanSpace", dimension, name). Usually an object of this class is not needed on its own. EuclideanSpace is the mother-class of the class Reals, which is generated automatically when a univariate absolutly continuous distribution is instantiated.

Slots

dimension

Object of class "numeric": the dimension of the space, by default = 1

name

Object of class "character": the name of the space, by default = "Euclidean Space"

Extends

Class "rSpace", directly.

Methods

initialize

signature(.Object = "EuclideanSpace"): initialize method

liesIn

signature(object = "EuclideanSpace", x = "numeric"): Does a particular vector lie in this space or not?

dimension

signature(object = "EuclideanSpace"): returns the dimension of the space

dimension<-

signature(object = "EuclideanSpace"): modifies the dimension of the space

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

rSpace-class Reals-class Distribution-class liesIn-methods

Examples

E <- EuclideanSpace(dimension = 2) 
dimension(E) # The dimension of this space is 2.
dimension(E) <- 3 # The dimension of this space is now 3.
liesIn(E,c(0,0,0)) # TRUE
liesIn(E,c(0,0)) # FALSE

Class "Exp"

Description

The exponential distribution with rate \lambda has density

f(x) = \lambda {e}^{- \lambda x}

for x \ge 0.

C.f. rexp

Objects from the Class

Objects can be created by calls of the form Exp(rate). This object is an exponential distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "ExpParameter": the parameter of this distribution (rate), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rexp)

d

Object of class "function": density function (calls function dexp)

p

Object of class "function": cumulative function (calls function pexp)

q

Object of class "function": inverse of the cumulative function (calls function qexp)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "ExpOrGammaOrChisq", directly.
Class "AbscontDistribution", by class "ExpOrGammaOrChisq".
Class "UnivariateDistribution", by class "AbscontDistribution". Class "Distribution", by class "AbscontDistribution".

Is-Relations

By means of setIs, R “knows” that a distribution object obj of class "Exp" also is a Gamma distribution with parameters shape = 1, scale = 1/rate(obj) and a Weibull distribution with parameters shape = 1, scale = 1/rate(obj)

Methods

initialize

signature(.Object = "Exp"): initialize method

rate

signature(object = "Exp"): returns the slot rate of the parameter of the distribution

rate<-

signature(object = "Exp"): modifies the slot rate of the parameter of the distribution

*

signature(e1 = "Exp", e2 = "numeric"): For the exponential distribution we use its closedness under positive scaling transformations.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

ExpParameter-class AbscontDistribution-class Reals-class rexp

Examples

E <- Exp(rate = 1) # E is a exp distribution with rate = 1.
r(E)(1) # one random number generated from this distribution, e.g. 0.4190765
d(E)(1) # Density of this distribution is 0.3678794 for x = 1.
p(E)(1) # Probability that x < 1 is 0.6321206.
q(E)(.1) # Probability that x < 0.1053605 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
rate(E) # rate of this distribution is 1.
rate(E) <- 2 # rate of this distribution is now 2.
is(E, "Gammad") # yes
as(E,"Gammad")
is(E, "Weibull") 
E+E+E ###  a Gammad -distribution
2*E+Gammad(scale=1)

Class "ExpOrGammaOrChisq"

Description

To have common methods, a class "ExpOrGammaOrChisq" is introduced as subclass of class "AbscontDistribution" and as superclass of classes "Chisq", "Exp", "Gammad". It is only used internally.

Objects from the Class

This class is virtual, hence cannot be instantiated.

Extends

Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "UnivariateDistribution".

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

Chisq-class Gammad-class Exp-class

Examples

## class only used internally

Class "ExpParameter"

Description

The parameter of an exponential distribution, used by Exp-class and DExp-class

Objects from the Class

Objects can be created by calls of the form new("ExpParameter", rate). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Exp is instantiated.

Slots

rate

Object of class "numeric": the rate of an exponential distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "ExpParameter"): initialize method

rate

signature(object = "ExpParameter"): returns the slot rate of the parameter of the distribution

rate<-

signature(object = "ExpParameter"): modifies the slot rate of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Exp-class DExp-class Parameter-class

Examples

W <- new("ExpParameter", rate = 1)
rate(W) # rate of this distribution is 1.
rate(W) <- 2 # rate of this distribution is now 2.

Class "FParameter"

Description

The parameter of a F distribution, used by Fd-class

Objects from the Class

Objects can be created by calls of the form new("FParameter", df1, df2, ncp). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Fd is instantiated.

Slots

df1

Object of class "numeric": the degrees of freedom of the nominator of an F distribution

df2

Object of class "numeric": the degrees of freedom of the denominator of an F distribution

ncp

Object of class "numeric": the noncentrality parameter of an F distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "FParameter"): initialize method

df1

signature(object = "FParameter"): returns the slot df1 of the parameter of the distribution

df1<-

signature(object = "FParameter"): modifies the slot df1 of the parameter of the distribution

df2

signature(object = "FParameter"): returns the slot df2 of the parameter of the distribution

df2<-

signature(object = "FParameter"): modifies the slot df2 of the parameter of the distribution

ncp

signature(object = "FParameter"): returns the slot ncp of the parameter of the distribution

ncp<-

signature(object = "FParameter"): modifies the slot ncp of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Fd-class Parameter-class

Examples

W <- new("FParameter", df1 = 1, df2 = 1, ncp = 0)
df2(W) # df2 of this distribution is 1.
df2(W) <- 2 # df2 of this distribution is now 2.

Class "Fd"

Description

The F distribution with df1 = n_1, by default = 1, and df2 = n_2, by default = 1, degrees of freedom has density

d(x) = \frac{\Gamma(n_1/2 + n_2/2)}{\Gamma(n_1/2)\Gamma(n_2/2)} \left(\frac{n_1}{n_2}\right)^{n_1/2} x^{n_1/2 -1} \left(1 + \frac{n_1 x}{n_2}\right)^{-(n_1 + n_2) / 2}%

for x > 0.

C.f. rf

Objects from the Class

Objects can be created by calls of the form Fd(df1, df2). This object is a F distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "FParameter": the parameter of this distribution (df1 and df2), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rf)

d

Object of class "function": density function (calls function df)

p

Object of class "function": cumulative function (calls function pf)

q

Object of class "function": inverse of the cumulative function (calls function qf)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Methods

initialize

signature(.Object = "Fd"): initialize method

df1

signature(object = "Fd"): returns the slot df1 of the parameter of the distribution

df1<-

signature(object = "Fd"): modifies the slot df1 of the parameter of the distribution

df2

signature(object = "Fd"): returns the slot df2 of the parameter of the distribution

df2<-

signature(object = "Fd"): modifies the slot df2 of the parameter of the distribution

Ad hoc methods

• An ad hoc method is provided for slot d if ncp!=0.

• For R Version <2.3.0 ad hoc methods are provided for slots q, r if ncp!=0; for R Version >=2.3.0 the methods from package stats are used.

Note

It is the distribution of the ratio of the mean squares of n1 and n2 independent standard normals, and hence of the ratio of two independent chi-squared variates each divided by its degrees of freedom. Since the ratio of a normal and the root mean-square of m independent normals has a Student's t_m distribution, the square of a t_m variate has a F distribution on 1 and m degrees of freedom.

The non-central F distribution is again the ratio of mean squares of independent normals of unit variance, but those in the numerator are allowed to have non-zero means and ncp is the sum of squares of the means.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

FParameter-class AbscontDistribution-class Reals-class rf

Examples

F <- Fd(df1 = 1, df2 = 1) # F is a F distribution with df=1 and df2=1.
r(F)(1) # one random number generated from this distribution, e.g. 29.37863
d(F)(1) # Density of this distribution is 0.1591549 for x=1 .
p(F)(1) # Probability that x<1 is 0.5.
q(F)(.1) # Probability that x<0.02508563 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
df1(F) # df1 of this distribution is 1.
df1(F) <- 2 # df1 of this distribution is now 2.
Fn <- Fd(df1 = 1, df2 = 1, ncp = 0.5) 
  # Fn is a F distribution with df=1, df2=1 and ncp =0.5.
d(Fn)(1) ## from R 2.3.0 on ncp no longer ignored...

Class "GammaParameter"

Description

The parameter of a gamma distribution, used by Gammad-class

Objects from the Class

Objects can be created by calls of the form new("GammaParameter", shape, scale). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Gammad is instantiated.

Slots

shape

Object of class "numeric": the shape of a Gamma distribution

scale

Object of class "numeric": the scale of a Gamma distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "GammaParameter"): initialize method

scale

signature(object = "GammaParameter"): returns the slot scale of a parameter of a Gamma distribution

scale<-

signature(object = "GammaParameter"): modifies the slot scale of a parameter of a Gamma distribution

shape

signature(object = "GammaParameter"): returns the slot shape of a parameter of a Gamma distribution

shape<-

signature(object = "GammaParameter"): modifies the slot shape of a parameter of a Gamma distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Gammad-class Parameter-class

Examples

W <- new("GammaParameter",scale=1,shape=1)
shape(W) # shape of this distribution is 1.
shape(W) <- 2 # shape of this distribution is now 2.

Class "Gammad"

Description

The Gammad distribution with parameters shape =\alpha, by default = 1, and scale =\sigma, by default = 1, has density

d(x)= \frac{1}{{\sigma}^{\alpha}\Gamma(\alpha)} {x}^{\alpha-1} e^{-x/\sigma}%

for x > 0, \alpha > 0 and \sigma > 0. The mean and variance are E(X) = \alpha\sigma and Var(X) = \alpha\sigma^2. C.f. rgamma

Objects from the Class

Objects can be created by calls of the form Gammad(scale, shape). This object is a gamma distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "GammaParameter": the parameter of this distribution (scale and shape), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rgamma)

d

Object of class "function": density function (calls function dgamma)

p

Object of class "function": cumulative function (calls function pgamma)

q

Object of class "function": inverse of the cumulative function (calls function qgamma)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "ExpOrGammaOrChisq", directly.
Class "AbscontDistribution", by class "ExpOrGammaOrChisq".
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "UnivariateDistribution".

Methods

initialize

signature(.Object = "Gammad"): initialize method

scale

signature(object = "Gammad"): returns the slot scale of the parameter of the distribution

scale<-

signature(object = "Gammad"): modifies the slot scale of the parameter of the distribution

shape

signature(object = "Gammad"): returns the slot shape of the parameter of the distribution

shape<-

signature(object = "Gammad"): modifies the slot shape of the parameter of the distribution

+

signature(e1 = "Gammad", e2 = "Gammad"): For the Gamma distribution we use its closedness under convolutions.

*

signature(e1 = "Gammad", e2 = "numeric"): For the Gamma distribution we use its closedness under positive scaling transformations.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

GammaParameter-class AbscontDistribution-class Reals-class rgamma

Examples

G <- Gammad(scale=1,shape=1) # G is a gamma distribution with scale=1 and shape=1.
r(G)(1) # one random number generated from this distribution, e.g. 0.1304441
d(G)(1) # Density of this distribution is 0.3678794 for x=1.
p(G)(1) # Probability that x<1 is 0.6321206.
q(G)(.1) # Probability that x<0.1053605 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
scale(G) # scale of this distribution is 1.
scale(G) <- 2 # scale of this distribution is now 2.

Class "Geom"

Description

The geometric distribution with prob = p has density

p(x) = p {(1-p)}^{x}

for x = 0, 1, 2, \ldots

C.f. rgeom

Objects from the Class

Objects can be created by calls of the form Geom(prob). This object is a geometric distribution.

Slots

img

Object of class "Naturals": The space of the image of this distribution has got dimension 1 and the name "Natural Space".

param

Object of class "NbinomParameter": the parameter of this distribution (prob), declared at its instantiation (size=1)

r

Object of class "function": generates random numbers (calls function rgeom)

d

Object of class "function": density function (calls function dgeom)

p

Object of class "function": cumulative function (calls function pgeom)

q

Object of class "function": inverse of the cumulative function (calls function qgeom). The quantile is defined as the smallest value x such that F(x) \ge p, where F is the distribution function.

support

Object of class "numeric": a (sorted) vector containing the support of the discrete density function

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "DiscreteDistribution", directly.
Class "Nbinom", directly.
Class "UnivariateDistribution", by class "DiscreteDistribution".
Class "Distribution", by class "DiscreteDistribution".

Contains-Relations

By means of a contains argument in the class declaration, R “knows” that a distribution object obj of class "Geom" also is a negative Binomial distribution with parameters size = 1, prob = prob(obj)

Methods

initialize

signature(.Object = "Geom"): initialize method

prob

signature(object = "Geom"): returns the slot prob of the parameter of the distribution

prob<-

signature(object = "Geom"): modifies the slot prob of the parameter of the distribution

Note

Working with a computer, we use a finite interval as support which carries at least mass 1-getdistrOption("TruncQuantile").

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Nbinom-class GeomParameter-class DiscreteDistribution-class Naturals-class rgeom

Examples

G <- Geom(prob = 0.5) # G is a geometric distribution with prob = 0.5.
r(G)(1) # one random number generated from this distribution, e.g. 0
d(G)(1) # Density of this distribution is 0.25 for x = 1.
p(G)(1) # Probability that x<1 is 0.75.
q(G)(.1) # x = 0 is the smallest value x such that p(G)(x) >= 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
prob(G) # prob of this distribution is 0.5.
prob(G) <- 0.6 # prob of this distribution is now 0.6.
as(G,"Nbinom")
G+G+G

Methods for function Huberize in Package ‘distr’

Description

Huberize-methods

Usage

Huberize(object, ...)
## S4 method for signature 'AcDcLcDistribution'
Huberize(object,lower,upper,
                    withSimplify = getdistrOption("simplifyD"))

Arguments

Value

the corresponding distribution of the truncated random variable

Methods

Huberize

signature(object = "AcDcLcDistribution"): returns the unconditioned distribution of min(upper,max(X,lower)), if X is distributed according to object; the result is of class "UnivarLebDecDistribution" in general.

See Also

Truncate

Examples

Hub <- Huberize(Norm(),lower=-1,upper=2)
Hub 
plot(Hub)

Class "Hyper"

Description

The hypergeometric distribution is used for sampling without replacement. The density of this distribution with parameters m, n and k (named Np, N-Np, and n, respectively in the reference below) is given by

p(x) = \left. {m \choose x}{n \choose k-x} \right/ {m+n \choose k}%

for x = 0, \ldots, k. C.f. rhyper

Objects from the Class

Objects can be created by calls of the form Hyper(m, n, k). This object is a hypergeometric distribution.

Slots

img

Object of class "Naturals": The space of the image of this distribution has got dimension 1 and the name "Natural Space".

param

Object of class "HyperParameter": the parameter of this distribution (m, n, k), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rhyper)

d

Object of class "function": density function (calls function dhyper)

p

Object of class "function": cumulative function (calls function phyper)

q

Object of class "function": inverse of the cumulative function (calls function qhyper). The \alpha-quantile is defined as the smallest value x such that p(x) \ge \alpha], where p is the cumulative function.

support:

Object of class "numeric": a (sorted) vector containing the support of the discrete density function

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "DiscreteDistribution", directly.
Class "UnivariateDistribution", by class "DiscreteDistribution".
Class "Distribution", by class "DiscreteDistribution".

Methods

initialize

signature(.Object = "Hyper"): initialize method

m

signature(object = "Hyper"): returns the slot m of the parameter of the distribution

m<-

signature(object = "Hyper"): modifies the slot m of the parameter of the distribution

n

signature(object = "Hyper"): returns the slot n of the parameter of the distribution

n<-

signature(object = "Hyper"): modifies the slot n of the parameter of the distribution

k

signature(object = "Hyper"): returns the slot k of the parameter of the distribution

k<-

signature(object = "Hyper"): modifies the slot k of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

HyperParameter-class DiscreteDistribution-class Naturals-class rhyper

Examples

H <- Hyper(m=3,n=3,k=3) # H is a hypergeometric distribution with m=3,n=3,k=3.
r(H)(1) # one random number generated from this distribution, e.g. 2
d(H)(1) # Density of this distribution is  0.45 for x=1.
p(H)(1) # Probability that x<1 is 0.5.
q(H)(.1) # x=1 is the smallest value x such that p(H)(x)>=0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
m(H) # m of this distribution is 3.
m(H) <- 2 # m of this distribution is now 2.

Class "HyperParameter"

Description

The parameter of a hypergeometric distribution, used by Hyper-class

Objects from the Class

Objects can be created by calls of the form new("HyperParameter", k, m, n). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Hyper is instantiated.

Slots

k

Object of class "numeric": k of a hypergeometric distribution

m

Object of class "numeric": m of a hypergeometric distribution

n

Object of class "numeric": n of a hypergeometric distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "HyperParameter"): initialize method

k

signature(object = "HyperParameter"): returns the slot k of the parameter of the distribution

k<-

signature(object = "HyperParameter"): modifies the slot k of the parameter of the distribution

m

signature(object = "HyperParameter"): returns the slot m of the parameter of the distribution

m<-

signature(object = "HyperParameter"): modifies the slot m of the parameter of the distribution

n

signature(object = "HyperParameter"): returns the slot n of the parameter of the distribution

n<-

signature(object = "HyperParameter"): modifies the slot n of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Hyper-class Parameter-class

Examples

W <- new("HyperParameter",k=3, m=3, n=3)
m(W) # m of this distribution is 3.
m(W) <- 2 # m of this distribution is now 2.

Internal Class "Integer"

Description

For the ease of method dispatch, there is an internal S4 class Integer, which is a subclass of numeric and has a straightforward validity method.

Objects from the Class

new("Integer",

Slots

.Data

Object of class "numeric"

Extends

Class "numeric", from data part. Class "vector", by class "numeric", distance 2.

Methods

coerce

signature(from = "numeric", to = "Integer"): create a "Integer" object from a "numeric" vector.

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

numeric, vector

Class unions in 'distr'

Description

Class unions in package distr defined for internal purposes; these are OptionalNumeric,

Details

These classes are used internally to make available methods or to allow slots of classes to be filled with varying types. In particular

• "OptionalNumeric": may contain objects of class "numeric" or "NULL"; it is used e.g. for slot nuisance of class "ParamFamParameter", as it may or may not be present but if so it has to be numeric.

Objects from the Class

All of these classes are virtual: No objects may be created from them.

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

numeric-class,

Class "Lattice"

Description

Class Lattice formalizes an affine linearly generated grid of (support) points pivot + (0:(Length-1)) * width; this is used for subclass LatticeDistribution of class DiscreteDistribution which in addition to the latter contains a slot lattice of class Lattice.

Usage

  Lattice(pivot = 0, width = 1, Length = 2, name = "a lattice")

Arguments

Objects from the Class

Objects may be generated by calling the generating function Lattice.

Slots

pivot

Object of class "numeric": — the pivot of the lattice; must be of length 1

width

Object of class "numeric": — the width of the lattice; must be of length 1 and must not be 0

Length

Object of class "numeric": — the width of the lattice; must be an integer > 0 of length 1

name

Object of class "character": the name of the space, by default = "a lattice"

Extends

Class "rSpace", directly.

Methods

pivot

signature(.Object = "Lattice"): returns the 'pivot' slot

pivot<-

signature(.Object = "Lattice"): modifies the 'pivot' slot

width

signature(.Object = "Lattice"): returns the 'width' slot

width<-

signature(.Object = "Lattice"): modifies the 'width' slot

Length

signature(.Object = "Lattice"): returns the 'Length' slot

Length<-

signature(.Object = "Lattice"): modifies the 'Length' slot

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

rSpace-class LatticeDistribution-class

Examples

L <- Lattice(pivot = 0, width = 1, Length = Inf, name = "the Naturals")
name(L)
pivot(L) <- 1 ### now starting from 1

Class "LatticeDistribution"

Description

The LatticeDistribution-class is the mother-class of the classes Binom, Dirac, Geom, Hyper, Nbinom and Poisson. It formalizes a distribution on a regular affine linear lattice.

Usage

  LatticeDistribution(lattice = NULL, supp = NULL, prob = NULL,
                       .withArith = FALSE, .withSim = FALSE,
                       DiscreteDistribution = NULL, check = TRUE,
                       Symmetry = NoSymmetry())

Arguments

Details

Typical usages are

  LatticeDistribution(DiscreteDistribution)
  LatticeDistribution(lattice, DiscreteDistribution)
  LatticeDistribution(lattice, supp, prob, .withArith, .withSim, check = FALSE)
  LatticeDistribution(lattice, supp, prob)
  LatticeDistribution(supp)
  

For the generating function LatticeDistribution(), the arguments are processed in the following order:
Arguments .withSim and .withArith are used in any case.
If there is an argument DiscreteDistribution (of the respective class), all its slots (except for .withSim and .withArith) will be used for filling the slots of the object of class LatticeDistribution()/AffLinLatticeDistribution(). If in addition, there is an argument lattice of class Lattice, it will be checked for consistency with argument DiscreteDistribution and if oK will be used for slot lattice of the object of class LatticeDistribution()/AffLinLatticeDistribution(). In case there is no lattice argument, slot lattice will be constructed from slot support from argument DiscreteDistribution.
If there is no argument DiscreteDistribution, but there are arguments supp and lattice (the latter of class Lattice) then these are checked for consistency and if oK, generating function DiscreteDistribution() is called with arguments supp, prob, .withArith, and .withSim to produce an object of class DiscreteDistribution the slots of which will be used for the filling the slots of the object of class LatticeDistribution()/AffLinLatticeDistribution(). If in this case, argument prob is not given explicitely, all elements in supp are equally weighted.
If there is no argument DiscreteDistribution, but there is an argument lattice of class Lattice (but no argument slot) then if Length(lattice) is finite, a corresponding support vector supp is generated from argument lattice and generating function DiscreteDistribution() is called with arguments supp, prob, .withArith, and .withSim to produce an object of class DiscreteDistribution the slots of which will be used for the filling the slots of the object of class LatticeDistribution(). If in the same situation Length(lattice) is not finite, a finite length for the support vector is extracted from argument prob and after generating supp one procedes as in the finite Length(lattice) case.
If there is no argument DiscreteDistribution and no argument lattice of class Lattice but an argument supp then it will be checked if supp makes for a lattice, and if so, DiscreteDistribution() is called with arguments supp, prob, .withArith, and .withSim to produce an object of class DiscreteDistribution the slots of which will be used for the filling the slots of the object of class LatticeDistribution(). The corresponding lattice-slot will be filled with information from argument supp.
The price for this flexibility of arguments, LatticeDistribution() may be called with, is that you should call LatticeDistribution() with named arguments only.
Note that internally we suppress lattice points from the support where the probability is 0.

Objects from the Class

The usual way to generate objects of class LatticeDistribution is to call the generating function LatticeDistribution() (see details).
Somewhat more flexible, but also proner to inconsistencies is a call to new("LatticeDistribution"), where you may explicitly specify random number generator, (counting) density, cumulative distribution and quantile functions. For conveniance, in this call to new("LatticeDistribution"), an additional possibility is to only specify the random number generator. The function RtoDPQ.d then approximates the three remaining slots d, p and q by random sampling.

Note

Working with a computer, we use a finite interval as support which carries at least mass 1-getdistrOption("TruncQuantile").

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

Parameter-class Lattice-class LatticeDistribution-class Reals-class RtoDPQ.d

Examples

  LatticeDistribution(DiscreteDistribution = DiscreteDistribution(supp =
                       c(4,3,2), prob=c(0.3,0.1,0.6)))
  LatticeDistribution(supp = c(4,3,2))

Class "LatticeDistribution"

Description

The LatticeDistribution-class is the mother-class of the classes Binom, Dirac, Geom, Hyper, Nbinom and Poisson. It formalizes a distribution on a regular affine linear lattice.

Objects from the Class

The usual way to generate objects of class LatticeDistribution is to call the generating function LatticeDistribution.
Somewhat more flexible, but also proner to inconsistencies is a call to new("LatticeDistribution"), where you may explicitly specify random number generator, (counting) density, cumulative distribution and quantile functions. For conveniance, in this call to new("LatticeDistribution"), an additional possibility is to only specify the random number generator. The function RtoDPQ.d then approximates the three remaining slots d, p and q by random sampling.

Slots

img

Object of class "Reals": the space of the image of this distribution which has dimension 1 and the name "Real Space"

param

Object of class "Parameter": the parameter of this distribution, having only the slot name "Parameter of a discrete distribution"

r

Object of class "function": generates random numbers

d

Object of class "function": (counting) density/probability function

p

Object of class "function": cumulative distribution function

q

Object of class "function": quantile function

support

Object of class "numeric": a (sorted) vector containing the support of the discrete density function

lattice

Object of class "Lattice": the lattice generating the support.

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "UnivariateDistribution", directly.
Class "Distribution", by class "UnivariateDistribution".

Methods

initialize

signature(.Object = "LatticeDistribution"): initialize method

-

signature(e1 = "LatticeDistribution"): application of ‘-’ to this lattice distribution

*

signature(e1 = "LatticeDistribution", e2 = "numeric"): multiplication of this lattice distribution by an object of class ‘numeric’

/

signature(e1 = "LatticeDistribution", e2 = "numeric"): division of this lattice distribution by an object of class ‘numeric’

+

signature(e1 = "LatticeDistribution", e2 = "numeric"): addition of this lattice distribution to an object of class ‘numeric’

-

signature(e1 = "LatticeDistribution", e2 = "numeric"): subtraction of an object of class ‘numeric’ from this lattice distribution

*

signature(e1 = "numeric", e2 = "LatticeDistribution"): multiplication of this lattice distribution by an object of class ‘numeric’

+

signature(e1 = "numeric", e2 = "LatticeDistribution"): addition of this lattice distribution to an object of class ‘numeric’

-

signature(e1 = "numeric", e2 = "LatticeDistribution"): subtraction of this lattice distribution from an object of class ‘numeric’

+

signature(e1 = "LatticeDistribution", e2 = "LatticeDistribution"): Convolution of two lattice distributions. Slots p, d and q are approximated by grids.

-

signature(e1 = "LatticeDistribution", e2 = "LatticeDistribution"): Convolution of two lattice distributions. The slots p, d and q are approximated by grids.

sqrt

signature(x = "LatticeDistribution"): exact image distribution of sqrt(x).

lattice

accessor method to the corresponding slot.

coerce

signature(from = "LatticeDistribution", to = "DiscreteDistribution"): coerces an object from "LatticeDistribution" to "DiscreteDistribution" thereby cancelling out support points with probability 0.

Internal subclass "AffLinLatticeDistribution"

To enhance accuracy of several functionals on distributions, mainly from package distrEx, there is an internally used (but exported) subclass "AffLinLatticeDistribution" which has extra slots a, b (both of class "numeric"), and X0 (of class "LatticeDistribution"), to capture the fact that the object has the same distribution as a * X0 + b. This is the class of the return value of methods

-

signature(e1 = "LatticeDistribution")

*

signature(e1 = "LatticeDistribution", e2 = "numeric")

/

signature(e1 = "LatticeDistribution", e2 = "numeric")

+

signature(e1 = "LatticeDistribution", e2 = "numeric")

-

signature(e1 = "LatticeDistribution", e2 = "numeric")

*

signature(e1 = "numeric", e2 = "LatticeDistribution")

+

signature(e1 = "numeric", e2 = "LatticeDistribution")

-

signature(e1 = "numeric", e2 = "LatticeDistribution")

-

signature(e1 = "AffLinLatticeDistribution")

*

signature(e1 = "AffLinLatticeDistribution", e2 = "numeric")

/

signature(e1 = "AffLinLatticeDistribution", e2 = "numeric")

+

signature(e1 = "AffLinLatticeDistribution", e2 = "numeric")

-

signature(e1 = "AffLinLatticeDistribution", e2 = "numeric")

*

signature(e1 = "numeric", e2 = "AffLinLatticeDistribution")

+

signature(e1 = "numeric", e2 = "AffLinLatticeDistribution")

-

signature(e1 = "numeric", e2 = "AffLinLatticeDistribution")

There is also an explicit coerce-method from class "AffLinLatticeDistribution" to class "AffLinDiscreteDistribution" which cancels out support points with probability 0.

Note

Working with a computer, we use a finite interval as support which carries at least mass 1-getdistrOption("TruncQuantile").

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

LatticeDistribution Parameter-class Lattice-class UnivariateDistribution-class DiscreteDistribution-class Binom-class Dirac-class Geom-class Hyper-class Nbinom-class Pois-class AbscontDistribution-class Reals-class RtoDPQ.d

Examples

B <- Binom(prob = 0.1,size = 10) # B is a Binomial distribution w/ prob=0.1 and size=10.
P <- Pois(lambda = 1) # P is a Poisson distribution with lambda = 1.
D1 <- B+1 # a new Lattice distributions with exact slots d, p, q
D2 <- D1*3 # a new Lattice distributions with exact slots d, p, q
D3 <- B+P # a new Lattice distributions with approximated slots d, p, q
D4 <- D1+P # a new Lattice distributions with approximated slots d, p, q
support(D4) # the (approximated) support of this distribution is 1, 2, ..., 21
r(D4)(1) # one random number generated from this distribution, e.g. 4
d(D4)(1) # The (approximated) density for x=1 is 0.1282716.
p(D4)(1) # The (approximated) probability that x<=1 is 0.1282716.
q(D4)(.5) # The (approximated) 50 percent quantile is 3.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)

Methods for Function Length in Package ‘distr’

Description

Length-methods

Methods

Length

signature(object = "Lattice"): returns the slot Length of the lattice

Length<-

signature(object = "Lattice"): modifies the slot Length of the lattice

Length

signature(object = "LatticeDistribution"): returns the slot Length of the lattice slot of the distribution

Length<-

signature(object = "LatticeDistribution"): modifies the slot Length of the lattice slot of the distribution

Class "Lnorm"

Description

The log normal distribution has density

d(x) = \frac{1}{\sqrt{2\pi}\sigma x} e^{-(\log(x) - \mu)^2/2 \sigma^2}%

where \mu, by default =0, and \sigma, by default =1, are the mean and standard deviation of the logarithm. C.f. rlnorm

Objects from the Class

Objects can be created by calls of the form Lnorm(meanlog, sdlog). This object is a log normal distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "LnormParameter": the parameter of this distribution (meanlog and sdlog), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rlnorm)

d

Object of class "function": density function (calls function dlnorm)

p

Object of class "function": cumulative function (calls function plnorm)

q

Object of class "function": inverse of the cumulative function (calls function qlnorm)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Methods

initialize

signature(.Object = "Lnorm"): initialize method

meanlog

signature(object = "Lnorm"): returns the slot meanlog of the parameter of the distribution

meanlog<-

signature(object = "Lnorm"): modifies the slot meanlog of the parameter of the distribution

sdlog

signature(object = "Lnorm"): returns the slot sdlog of the parameter of the distribution

sdlog<-

signature(object = "Lnorm"): modifies the slot sdlog of the parameter of the distribution

*

signature(e1 = "Lnorm", e2 = "numeric"): For the Lognormal distribution we use its closedness under positive scaling transformations.

Note

The mean is E(X) = exp(\mu + 1/2 \sigma^2), and the variance Var(X) = exp(2\mu + \sigma^2)(exp(\sigma^2) - 1) and hence the coefficient of variation is \sqrt{exp(\sigma^2) - 1} which is approximately \sigma when that is small (e.g., \sigma < 1/2).

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

LnormParameter-class AbscontDistribution-class Reals-class rlnorm

Examples

L <- Lnorm(meanlog=1,sdlog=1) # L is a lnorm distribution with mean=1 and sd=1.
r(L)(1) # one random number generated from this distribution, e.g. 3.608011
d(L)(1) # Density of this distribution is 0.2419707 for x=1.
p(L)(1) # Probability that x<1 is 0.1586553.
q(L)(.1) # Probability that x<0.754612 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
meanlog(L) # meanlog of this distribution is 1.
meanlog(L) <- 2 # meanlog of this distribution is now 2.

Class "LnormParameter"

Description

The parameter of a log normal distribution, used by Lnorm-class

Objects from the Class

Objects can be created by calls of the form new("LnormParameter", meanlog, sdlog). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Lnorm is instantiated.

Slots

meanlog

Object of class "numeric": the mean of a log normal distribution

sdlog

Object of class "numeric": the sd of a log normal distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "LnormParameter"): initialize method

sdlog

signature(object = "LnormParameter"): returns the slot sdlog of the parameter of the distribution

sdlog<-

signature(object = "LnormParameter"): modifies the slot sdlog of the parameter of the distribution

meanlog

signature(object = "LnormParameter"): returns the slot meanlog of the parameter of the distribution

meanlog<-

signature(object = "LnormParameter"): modifies the slot meanlog of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Lnorm-class Parameter-class

Examples

W <- new("LnormParameter",sdlog=1,meanlog=0)
meanlog(W) # meanlog of this distribution is 0.
meanlog(W) <- 2 # meanlog of this distribution is now 2.

Class "Logis"

Description

The Logistic distribution with location = \mu, by default = 0, and scale = \sigma, by default = 1, has distribution function

p(x) = \frac{1}{1 + e^{-(x-\mu)/\sigma}}%

and density

d(x)= \frac{1}{\sigma}\frac{e^{(x-\mu)/\sigma}}{(1 + e^{(x-\mu)/\sigma})^2}%

It is a long-tailed distribution with mean \mu and variance \pi^2/3 \sigma^2. C.f. rlogis

Objects from the Class

Objects can be created by calls of the form Logis(location, scale). This object is a logistic distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "LogisParameter": the parameter of this distribution (location and scale), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rlogis)

d

Object of class "function": density function (calls function dlogis)

p

Object of class "function": cumulative function (calls function plogis)

q

Object of class "function": inverse of the cumulative function (calls function qlogis)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Methods

initialize

signature(.Object = "Logis"): initialize method

location

signature(object = "Logis"): returns the slot location of the parameter of the distribution

location<-

signature(object = "Logis"): modifies the slot location of the parameter of the distribution

scale

signature(object = "Logis"): returns the slot scale of the parameter of the distribution

scale<-

signature(object = "Logis"): modifies the slot scale of the parameter of the distribution

*

signature(e1 = "Logis", e2 = "numeric")

+

signature(e1 = "Logis", e2 = "numeric"): For the logistic location scale family we use its closedness under affine linear transformations.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

LogisParameter-class AbscontDistribution-class Reals-class rlogis

Examples

L <- Logis(location = 1,scale = 1)
# L is a logistic distribution with  location = 1 and scale = 1.
r(L)(1) # one random number generated from this distribution, e.g. 5.87557
d(L)(1) # Density of this distribution is 0.25 for x = 1.
p(L)(1) # Probability that x < 1 is 0.5.
q(L)(.1) # Probability that x < -1.197225 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
location(L) # location of this distribution is 1.
location(L) <- 2 # location of this distribution is now 2.

Class "LogisParameter"

Description

The parameter of a logistic distribution, used by Logis-class

Objects from the Class

Objects can be created by calls of the form new("LogisParameter", scale, location). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Logis is instantiated.

Slots

scale

Object of class "numeric": the scale of a logistic distribution

location

Object of class "numeric": the location of a logistic distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "LogisParameter"): initialize method

location

signature(object = "LogisParameter"): returns the slot location of the parameter of the distribution

location<-

signature(object = "LogisParameter"): modifies the slot location of the parameter of the distribution

scale

signature(object = "LogisParameter"): returns the slot scale of the parameter of the distribution

scale<-

signature(object = "LogisParameter"): modifies the slot scale of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Logis-class Parameter-class

Examples

W <- new("LogisParameter",location=0,scale=1)
scale(W) # scale of this distribution is 1.
scale(W) <- 2 # scale of this distribution is now 2.

Methods for Functions from group ‘Math’ in Package ‘distr’

Description

Math-methods provide automatical generation of image distributions for random variables transformed by functions from group Math

Methods

Math

signature(x = "AbscontDistribution"): application of a mathematical function from group Math, e.g. sin or exp (including log, log10, gamma, lgamma, digamma), to this absolutely continouos distribution

Math

signature(x = "DiscreteDistribution"): application of a mathematical function, e.g. sin or exp (including log, log10, gamma, lgamma, digamma), to this discrete distribution

Math

signature(x = "UnivarLebDecDistribution"): application of a mathematical function from group Math, e.g. sin or exp (including log, log10, gamma, lgamma), to this Lebesgue decomposed distribution

Math

signature(x = "UnivarLebDecDistribution"): application of a mathematical function from group Math, e.g. sin or exp (including log, log10, gamma, lgamma), to this distribution of class "AcDcLcDistribution"

abs

signature(x = "AbscontDistribution"): application of function abs to this absolutely continouos distribution; (exactly)

abs

signature(x = "DiscreteDistribution"): application of function abs to this discrete distribution; (exactly)

sign

signature(x = "AbscontDistribution"): application of function abs to this absolutely continouos distribution; (exactly)

sign

signature(x = "DiscreteDistribution"): application of function abs to this discrete continouos distribution; (exactly)

exp

signature(x = "AbscontDistribution"): application of function exp to this absolutely continouos distribution; (exactly)

exp

signature(x = "DiscreteDistribution"): application of function exp to this discrete distribution; (exactly)

log

signature(x = "AbscontDistribution"): application of function log to this absolutely continouos distribution; (exactly for R-version >2.5.1)

log

signature(x = "DiscreteDistribution"): application of function log to this discrete distribution; (exactly for R-version >2.5.1)

Methods for Function Max in Package ‘distr’

Description

Max-methods

Methods

Max

signature(object = "UnifParameter"): returns the slot Max of the parameter of the distribution

Max<-

signature(object = "UnifParameter"): modifies the slot Max of the parameter of the distribution

Max

signature(object = "Unif"): returns the slot Max of the parameter of the distribution

Max<-

signature(object = "Unif"): modifies the slot Max of the parameter of the distribution

Methods for Function Min in Package ‘distr’

Description

Min-methods

Methods

Min

signature(object = "UnifParameter"): returns the slot Min of the parameter of the distribution

Min<-

signature(object = "UnifParameter"): modifies the slot Min of the parameter of the distribution

Min

signature(object = "Unif"): returns the slot Min of the parameter of the distribution

Min<-

signature(object = "Unif"): modifies the slot Min of the parameter of the distribution

Methods for functions Minimum and Maximum in Package ‘distr’

Description

Minimum and Maximum-methods

Usage

Minimum(e1, e2, ...)
Maximum(e1, e2, ...) 
## S4 method for signature 'AbscontDistribution,AbscontDistribution'
Minimum(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,DiscreteDistribution'
Minimum(e1,e2, ...)
## S4 method for signature 'AbscontDistribution,Dirac'
Minimum(e1,e2, 
                   withSimplify = getdistrOption("simplifyD"))
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
Minimum(e1,e2, 
                   withSimplify = getdistrOption("simplifyD"))
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
Maximum(e1,e2, 
                   withSimplify = getdistrOption("simplifyD"))
## S4 method for signature 'AbscontDistribution,numeric'
Minimum(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,numeric'
Minimum(e1,e2, ...)
## S4 method for signature 'AcDcLcDistribution,numeric'
Minimum(e1,e2,
                   withSimplify = getdistrOption("simplifyD"))
## S4 method for signature 'AcDcLcDistribution,numeric'
Maximum(e1,e2, 
                   withSimplify = getdistrOption("simplifyD"))

Arguments

Value

the corresponding distribution of the minimum / maximum

Methods

Minimum

signature(e1 = "AbscontDistribution", e2 = "AbscontDistribution"): returns the distribution of min(X1,X2), if X1,X2 are independent and distributed according to e1 and e2 respectively; the result is again of class "AbscontDistribution"

Minimum

signature(e1 = "DiscreteDistribution", e2 = "DiscreteDistribution"): returns the distribution of min(X1,X2), if X1,X2 are independent and distributed according to e1 and e2 respectively; the result is again of class "DiscreteDistribution"

Minimum

signature(e1 = "AbscontDistribution", e2 = "Dirac"): returns the distribution of min(X1,X2), if X1,X2 are distributed according to e1 and e2 respectively; the result is of class "UnivarLebDecDistribution"

Minimum

signature(e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution"): returns the distribution of min(X1,X2), if X1,X2 are distributed according to e1 and e2 respectively; the result is of class "UnivarLebDecDistribution"

Minimum

signature(e1 = "AcDcLcDistribution", e2 = "numeric"): if e2 = n, returns the distribution of min(X1,X2,...,Xn), if X1,X2, ..., Xn are i.i.d. according to e1; the result is of class "UnivarLebDecDistribution"

Maximum

signature(e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution"): returns the distribution of max(X1,X2), if X1,X2 are distributed according to e1 and e2 respectively; translates into -Minimum(-e1,-e2); the result is of class "UnivarLebDecDistribution"

Maximum

signature(e1 = "AcDcLcDistribution", e2 = "numeric"): if e2 = n, returns the distribution of max(X1,X2,...,Xn), if X1,X2, ..., Xn are i.i.d. according to e1; translates into -Minimum(-e1,e2); the result is of class "UnivarLebDecDistribution"

See Also

Huberize, Truncate

Examples

## IGNORE_RDIFF_BEGIN
plot(Maximum(Unif(0,1), Minimum(Unif(0,1), Unif(0,1))))
plot(Minimum(Exp(4),4))
## IGNORE_RDIFF_END


## a sometimes lengthy example...
plot(Minimum(Norm(),Pois()))

Class "Naturals"

Description

The distribution-classes contain a slot where the sample space is stored. Typically, discrete random variables take naturals as values.

Usage

Naturals()

Objects from the Class

Objects could theoretically be created by calls of the form new("Naturals", dimension, name). Usually an object of this class is not needed on its own. It is generated automatically when a univariate discrete distribution is instantiated.

Slots

dimension

Object of class "character": the dimension of the space, by default = 1

name

Object of class "character": the name of the space, by default = "Natural Space"

Extends

Class "Reals", directly.
Class "EuclideanSpace", by class "Reals".
Class "rSpace", by class "Reals".

Methods

initialize

signature(.Object = "Naturals"): initialize method

liesIn

signature(object = "Naturals", x = "numeric"): Does a particular vector only contain naturals?

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Reals-class DiscreteDistribution-class

Examples

N <- Naturals()
liesIn(N,1) # TRUE
liesIn(N,c(0,1)) # FALSE
liesIn(N,0.1) # FALSE

Class "Nbinom"

Description

The negative binomial distribution with size = n, by default =1, and prob = p, by default =0.5, has density

d(x) = \frac{\Gamma(x+n)}{\Gamma(n) x!} p^n (1-p)^x

for x = 0, 1, 2, \ldots

This represents the number of failures which occur in a sequence of Bernoulli trials before a target number of successes is reached. C.f. rnbinom

Objects from the Class

Objects can be created by calls of the form Nbinom(prob, size). This object is a negative binomial distribution.

Slots

img

Object of class "Naturals": The space of the image of this distribution has got dimension 1 and the name "Natural Space".

param

Object of class "NbinomParameter": the parameter of this distribution (prob, size), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rnbinom)

d

Object of class "function": density function (calls function dnbinom)

p

Object of class "function": cumulative function (calls function pnbinom)

q

Object of class "function": inverse of the cumulative function (calls function qnbinom). The quantile is defined as the smallest value x such that F(x) \ge p, where F is the distribution function.

support

Object of class "numeric": a (sorted) vector containing the support of the discrete density function

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "DiscreteDistribution", directly.
Class "UnivariateDistribution", by class "DiscreteDistribution".
Class "Distribution", by class "DiscreteDistribution".

Methods

initialize

signature(.Object = "Nbinom"): initialize method

prob

signature(object = "Nbinom"): returns the slot prob of the parameter of the distribution

prob<-

signature(object = "Nbinom"): modifies the slot prob of the parameter of the distribution

size

signature(object = "Nbinom"): returns the slot size of the parameter of the distribution

size<-

signature(object = "Nbinom"): modifies the slot size of the parameter of the distribution

+

signature(e1 = "Nbinom", e2 = "Nbinom"): For the negative binomial distribution we use its closedness under convolutions.

Note

Working with a computer, we use a finite interval as support which carries at least mass 1-getdistrOption("TruncQuantile").

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

NbinomParameter-class Geom-class DiscreteDistribution-class Naturals-class rnbinom

Examples

N <- Nbinom(prob = 0.5, size = 1) # N is a binomial distribution with prob=0.5 and size=1.
r(N)(1) # one random number generated from this distribution, e.g. 3
d(N)(1) # Density of this distribution is  0.25 for x=1.
p(N)(0.4) # Probability that x<0.4 is 0.5.
q(N)(.1) # x=0 is the smallest value x such that p(B)(x)>=0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
size(N) # size of this distribution is 1.
size(N) <- 2 # size of this distribution is now 2.

Class "NbinomParameter"

Description

The parameter of a negative binomial distribution, used by Nbinom-class

Objects from the Class

Objects can be created by calls of the form new("NbinomParameter", prob, size). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Nbinom is prepared.

Slots

prob

Object of class "numeric": the probability of a negative binomial distribution

size

Object of class "numeric": the size of a negative binomial distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "NbinomParameter"): initialize method

prob

signature(object = "NbinomParameter"): returns the slot prob of the parameter of the distribution

prob<-

signature(object = "NbinomParameter"): modifies the slot prob of the parameter of the distribution

size

signature(object = "NbinomParameter"): returns the slot size of the parameter of the distribution

size<-

signature(object = "NbinomParameter"): modifies the slot size of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Nbinom-class Parameter-class

Examples

W <- new("NbinomParameter",prob=0.5,size=1)
size(W) # size of this distribution is 1.
size(W) <- 2 # size of this distribution is now 2.

Generating function for NoSymmetry-class

Description

Generates an object of class "NoSymmetry".

Usage

NoSymmetry()

Value

Object of class "NoSymmetry"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

NoSymmetry-class, DistributionSymmetry-class

Examples

NoSymmetry()

## The function is currently defined as
function(){ new("NoSymmetry") }

Class for Non-symmetric Distributions

Description

Class for non-symmetric distributions.

Objects from the Class

Objects can be created by calls of the form new("NoSymmetry"). More frequently they are created via the generating function NoSymmetry.

Slots

type

Object of class "character": contains “non-symmetric distribution”

SymmCenter

Object of class "NULL"

Extends

Class "DistributionSymmetry", directly.
Class "Symmetry", by class "DistributionSymmetry".

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

NoSymmetry, Distribution-class

Examples

new("NoSymmetry")

Class "Norm"

Description

The normal distribution has density

f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-\mu)^2/2\sigma^2}

where \mu is the mean of the distribution and \sigma the standard deviation. C.f. rnorm

Objects from the Class

Objects can be created by calls of the form Norm(mean, sd). This object is a normal distribution.

Slots

img

Object of class "Reals": The domain of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "UniNormParameter": the parameter of this distribution (mean and sd), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rnorm)

d

Object of class "function": density function (calls function dnorm)

p

Object of class "function": cumulative function (calls function pnorm)

q

Object of class "function": inverse of the cumulative function (calls function qnorm)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Methods

-

signature(e1 = "Norm", e2 = "Norm")

+

signature(e1 = "Norm", e2 = "Norm"): For the normal distribution the exact convolution formulas are implemented thereby improving the general numerical approximation.

*

signature(e1 = "Norm", e2 = "numeric")

+

signature(e1 = "Norm", e2 = "numeric"): For the normal distribution we use its closedness under affine linear transformations.

initialize

signature(.Object = "Norm"): initialize method

mean

signature(object = "Norm"): returns the slot mean of the parameter of the distribution

mean<-

signature(object = "Norm"): modifies the slot mean of the parameter of the distribution

sd

signature(object = "Norm"): returns the slot sd of the parameter of the distribution

sd<-

signature(object = "Norm"): modifies the slot sd of the parameter of the distribution

further arithmetic methods see operators-methods

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

UniNormParameter-class AbscontDistribution-class Reals-class rnorm

Examples

N <- Norm(mean=1,sd=1) # N is a normal distribution with mean=1 and sd=1.
r(N)(1) # one random number generated from this distribution, e.g. 2.257783
d(N)(1) # Density of this distribution is  0.3989423 for x=1.
p(N)(1) # Probability that x<1 is 0.5.
q(N)(.1) # Probability that x<-0.2815516 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
mean(N) # mean of this distribution is 1.
sd(N) <- 2 # sd of this distribution is now 2.
M <- Norm() # M is a normal distribution with mean=0 and sd=1.
O <- M+N # O is a normal distribution with mean=1 (=1+0) and sd=sqrt(5) (=sqrt(2^2+1^2)).

Class "NormParameter"

Description

The parameter of a normal distribution, used by Norm-class

Objects from the Class

Objects can be created by calls of the form new("NormParameter", sd, mean). Usually an object of this class is not needed on its own. It is the mother-class of the class UniNormParameter, which is generated automatically when such a distribution is instantiated.

Slots

sd

Object of class "numeric": the sd of a normal distribution

mean

Object of class "numeric": the mean of a normal distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "NormParameter"): initialize method

mean

signature(object = "NormParameter"): returns the slot mean of the parameter of the distribution

mean<-

signature(object = "NormParameter"): modifies the slot mean of the parameter of the distribution

sd

signature(object = "NormParameter"): returns the slot sd of the parameter of the distribution

sd<-

signature(object = "NormParameter"): modifies the slot sd of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Norm-class Parameter-class

Examples

W <- new("NormParameter", mean = 0, sd = 1)
sd(W) # sd of this distribution is 1.
sd(W) <- 2 # sd of this distribution is now 2.

Classes "OptionalParameter", "OptionalMatrix"

Description

auxiliary classes; may contain either a Parameter or NULL, resp. a matrix or NULL cf. J. Chambers, "green book".

Objects from the Class

"OptionalParameter" is a virtual Class: No objects may be created from it; "OptionalMatrix" is a class generated by setClassUnion() so may contain NULL or any matrix

Methods

No methods defined with class "OptionalParameter" in the signature.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Parameter-class, AbscontDistribution-class

Class "Parameter"

Description

Parameter is the mother-class of all Parameter classes.

Objects from the Class

Objects can be created by calls of the form new("Parameter").

Slots

name

Object of class "character": a name / comment for the parameters

Methods

name

signature(object = "Parameter"): returns the name of the parameter

name<-

signature(object = "Parameter"): modifies the name of the parameter

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Distribution-class

Class "Pois"

Description

The Poisson distribution has density

p(x) = \frac{\lambda^x e^{-\lambda}}{x!}

for x = 0, 1, 2, \ldots. The mean and variance are E(X) = Var(X) = \lambda.

C.f. rpois

Objects from the Class

Objects can be created by calls of the form Pois(lambda). This object is a Poisson distribution.

Slots

img

Object of class "Naturals": The space of the image of this distribution has got dimension 1 and the name "Natural Space".

param

Object of class "PoisParameter": the parameter of this distribution (lambda), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rpois)

d

Object of class "function": density function (calls function dpois)

p

Object of class "function": cumulative function (calls function ppois)

q

Object of class "function": inverse of the cumulative function (calls function qpois). The quantile is defined as the smallest value x such that F(x) \ge p, where F is the distribution function.

support

Object of class "numeric": a (sorted) vector containing the support of the discrete density function

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "DiscreteDistribution", directly. Class "UnivariateDistribution", by class "DiscreteDistribution". Class "Distribution", by class "DiscreteDistribution".

Methods

+

signature(e1 = "Pois", e2 = "Pois"): For the Poisson distribution the exact convolution formula is implemented thereby improving the general numerical approximation.

initialize

signature(.Object = "Pois"): initialize method

lambda

signature(object = "Pois"): returns the slot lambda of the parameter of the distribution

lambda<-

signature(object = "Pois"): modifies the slot lambda of the parameter of the distribution

Note

Working with a computer, we use a finite interval as support which carries at least mass 1-getdistrOption("TruncQuantile").

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

PoisParameter-class DiscreteDistribution-class Naturals-class rpois

Examples

P <- Pois(lambda = 1) # P is a Poisson distribution with lambda = 1.
r(P)(1) # one random number generated from this distribution, e.g. 1
d(P)(1) # Density of this distribution is 0.3678794 for x = 1.
p(P)(0.4) # Probability that x < 0.4 is 0.3678794.
q(P)(.1) # x = 0 is the smallest value x such that p(B)(x) >= 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
lambda(P) # lambda of this distribution is 1.
lambda(P) <- 2 # lambda of this distribution is now 2.
R <- Pois(lambda = 3) # R is a Poisson distribution with lambda = 2.
S <- P + R # R is a Poisson distribution with lambda = 5(=2+3).

Class "PoisParameter"

Description

The parameter of a Poisson distribution, used by Pois-class

Objects from the Class

Objects can be created by calls of the form new("PoisParameter", lambda). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Pois is prepared.

Slots

lambda

Object of class "numeric": the lambda of a Poisson distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "PoisParameter"): initialize method

lambda

signature(object = "PoisParameter"): returns the slot lambda of the parameter of the distribution

lambda<-

signature(object = "PoisParameter"): modifies the slot lambda of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Pois-class Parameter-class

Examples

W <- new("PoisParameter",lambda = 1)
lambda(W) # lambda of this distribution is 1.
lambda(W) <- 2 # lambda of this distribution is now 2.

Generating functions for PosSemDefSymmMatrix-class resp. PosDefSymmMatrix-class

Description

Generates an object of class "PosSemDefSymmMatrix" resp. of class "PosDefSymmMatrix".

Usage

PosSemDefSymmMatrix(mat)
       PosDefSymmMatrix(mat)

Arguments

Details

If mat is no matrix, as.matrix is applied.

Value

Object of class "PosSemDefSymmMatrix" resp. of class "PosDefSymmMatrix"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de, Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

PosDefSymmMatrix-class

Examples

PosSemDefSymmMatrix(1)
PosSemDefSymmMatrix(diag(2))
PosDefSymmMatrix(1)
PosDefSymmMatrix(diag(2))

Positive-[Semi-]definite, symmetric matrices

Description

The class of positive-[semi-]definite, symmetric matrices.

Objects from the Class

Objects can be created by calls of the form new("PosSemDefSymmMatrix", ...) resp. new("PosDefSymmMatrix", ...). More frequently they are created via the generating functions PosSemDefSymmMatrix resp. PosDefSymmMatrix.

Slots

.Data

Object of class "matrix". A numeric matrix with finite entries.

Extends

[Class "PosSemDefSymmMatrix", directly] Class "matrix", from data part.
Class "structure", by class "matrix".
Class "array", by class "matrix".
Class "vector", by class "matrix", with explicit coerce.
Class "vector", by class "matrix", with explicit coerce.

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de, Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

PosDefSymmMatrix, matrix-class

Examples

new("PosDefSymmMatrix", diag(2))

Class "Reals"

Description

Particular case of a one-dimensional Euclidean Space

Usage

Reals()

Objects from the Class

Objects could theoretically be created by calls of the form new("Reals", dimension, name). Usually an object of this class is not needed on its own. It is generated automatically when a univariate absolutly continuous distribution is instantiated.

Slots

dimension

Object of class "character": the dimension of the space, by default = 1

name

Object of class "character": the name of the space, by default = "Real Space"

Extends

Class "EuclideanSpace", directly.
Class "rSpace", by class "EuclideanSpace".

Methods

initialize

signature(.Object = "Reals"): initialize method

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

EuclideanSpace-class Naturals-class AbscontDistribution-class

Examples

R <- Reals()
liesIn(R,c(0,0)) # FALSE

Default procedure to fill slots d,p,q given r for a.c. distributions

Description

function to do get empirical density, cumulative distribution and quantile function from random numbers

Usage

RtoDPQ(r, e = getdistrOption("RtoDPQ.e"), 
       n = getdistrOption("DefaultNrGridPoints"), y = NULL)

Arguments

Details

RtoDPQ generates 10^e random numbers, by default

e = RtoDPQ.e

. Instead of using simulated grid points, we have an optional parameter y for using N. Horbenko's quantile trick: i.e.; on an equally spaced grid x.grid on [0,1], apply f(q(x)(x.grid)) and write the result to y and produce density and cdf from this value y given to RtoDPQ as argument (instead of simulating grid points).

The density is formed on the basis of n points using approxfun and density, by default

n = DefaultNrGridPoints

. The cumulative distribution function and the quantile function are also created on the basis of n points using approxfun and ecdf. Of course, the results are usually not exact as they rely on random numbers.

Value

RtoDPQ returns a list of functions.

Note

Use RtoDPQ for absolutely continuous and RtoDPQ.d for discrete distributions.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

UnivariateDistribution-class, density, approxfun, ecdf

Examples

set.seed(20230508)
rn2 <- function(n){rnorm(n)^2}
x <- RtoDPQ(r = rn2, e = 4, n = 512)
# returns density, cumulative distribution and quantile function of
# squared standard normal distribution
## IGNORE_RDIFF_BEGIN
x$dfun(4)
RtoDPQ(r = rn2, e = 5, n = 1024) # for a better result
## IGNORE_RDIFF_END
rp2 <- function(n){rpois(n, lambda = 1)^2}
x <- RtoDPQ.d(r = rp2, e = 5)
# returns density, cumulative distribution and quantile function of
# squared Poisson distribution with parameter lambda=1

Default procedure to fill slots d,p,q given r for Lebesgue decomposed distributions

Description

function to do get empirical density, cumulative distribution and quantile function from random numbers

Usage

RtoDPQ.LC(r, e = getdistrOption("RtoDPQ.e"), 
          n = getdistrOption("DefaultNrGridPoints"), y = NULL)

Arguments

Details

RtoDPQ.LC generates 10^e random numbers, by default

e = RtoDPQ.e

. Replicates are assumed to be part of the discrete part, unique values to be part of the a.c. part of the distribution. For the replicated ones, we generate a discrete distribution by a call to DiscreteDistribution.

For the a.c. part, similarly to RtoDPQ we have an optional parameter y for using N. Horbenko's quantile trick: i.e.; on an equally spaced grid x.grid on [0,1], apply f(q(x)(x.grid)), write the result to y and use these values instead of simulated ones.

The a.c. density is formed on the basis of n points using approxfun and density (applied to the unique values), by default

n = DefaultNrGridPoints

. The cumulative distribution function is based on all random variables, and, as well as the quantile function, is also created on the basis of n points using approxfun and ecdf. Of course, the results are usually not exact as they rely on random numbers.

Value

RtoDPQ.LC returns an object of class UnivarLebDecDistribution.

Note

Use RtoDPQ for absolutely continuous and RtoDPQ.d for discrete distributions.

Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

See Also

UnivariateDistribution-class, density, approxfun, ecdf

Examples

set.seed(20230508)
rn2 <- function(n)ifelse(rbinom(n,1,0.3),rnorm(n)^2,rbinom(n,4,.3))
x <- RtoDPQ.LC(r = rn2, e = 4, n = 512)
plot(x)
# returns density, cumulative distribution and quantile function of
# squared standard normal distribution
## IGNORE_RDIFF_BEGIN
d.discrete(x)(4)
## IGNORE_RDIFF_END
x2 <- RtoDPQ.LC(r = rn2, e = 5, n = 1024) # for a better result
plot(x2)

Default procedure to fill slots d,p,q given r for discrete distributions

Description

function to do get empirical density, cumulative distribution and quantile function from random numbers

Usage

RtoDPQ.d(r, e = getdistrOption("RtoDPQ.e"))

Arguments

Details

RtoDPQ.d generates 10^e random numbers, by default e = RtoDPQ.e which are used to produce a density, cdf and quantile function. Of course, the results are usually not exact as they rely on random numbers.

Value

RtoDPQ returns a list of functions.

Note

Use RtoDPQ for absolutely continuous and RtoDPQ.d for discrete distributions.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

UnivariateDistribution-class, density, approxfun, ecdf

Examples

set.seed(20230508)
rn2 <- function(n){rnorm(n)^2}
x <- RtoDPQ(r = rn2, e = 4, n = 512)
# returns density, cumulative distribution and quantile function of
# squared standard  normal distribution
## IGNORE_RDIFF_BEGIN
x$dfun(4)
RtoDPQ(r = rn2, e = 5, n = 1024) # for a better result
## IGNORE_RDIFF_END
rp2 <- function(n){rpois(n, lambda = 1)^2}
x <- RtoDPQ.d(r = rp2, e = 5)
# returns density, cumulative distribution and quantile function of
# squared Poisson distribution with parameter lambda=1

Generating function for SphericalSymmetry-class

Description

Generates an object of class "SphericalSymmetry".

Usage

SphericalSymmetry(SymmCenter = 0)

Arguments

Value

Object of class "SphericalSymmetry"

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

SphericalSymmetry-class, DistributionSymmetry-class

Examples

SphericalSymmetry()

## The function is currently defined as
function(SymmCenter = 0){ 
    new("SphericalSymmetry", SymmCenter = SymmCenter) 
}

Class for Spherical Symmetric Distributions

Description

Class for spherical symmetric distributions.

Objects from the Class

Objects can be created by calls of the form new("SphericalSymmetry"). More frequently they are created via the generating function SphericalSymmetry. Spherical symmetry for instance leads to a simplification for the computation of optimally robust influence curves.

Slots

type

Object of class "character": contains “spherical symmetric distribution”

SymmCenter

Object of class "numeric": center of symmetry

Extends

Class "EllipticalSymmetry", directly.
Class "DistributionSymmetry", by class "EllipticalSymmetry".
Class "Symmetry", by class "EllipticalSymmetry".

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

SphericalSymmetry, DistributionSymmetry-class

Examples

new("SphericalSymmetry")

Class of Symmetries

Description

Class of symmetries of various objects.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

type

Object of class "character": discribes type of symmetry.

SymmCenter

Object of class "ANY": center of symmetry.

Methods

type

signature(object = "Symmetry"): accessor function for slot type

SymmCenter

signature(object = "Symmetry"): accessor function for slot SymmCenter

show

signature(object = "Symmetry")

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

See Also

DistributionSymmetry-class, OptionalNumeric-class

Class "TParameter"

Description

The parameter of a t distribution, used by Td-class

Objects from the Class

Objects can be created by calls of the form new("TParameter", df, ncp). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Td is instantiated.

Slots

df

Object of class "numeric": the degrees of freedom of a T distribution

ncp

Object of class "numeric": the noncentrality parameter of a T distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "TParameter"): initialize method

df

signature(object = "TParameter"): returns the slot df of the parameter of the distribution

df<-

signature(object = "TParameter"): modifies the slot df of the parameter of the distribution

ncp

signature(object = "TParameter"): returns the slot ncp of the parameter of the distribution

ncp<-

signature(object = "TParameter"): modifies the slot ncp of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Td-class Parameter-class

Examples

W <- new("TParameter",df=1, ncp = 0)
df(W) # df of this distribution is 1.
df(W) <- 2 # df of this distribution is now 2.

Class "Td"

Description

The t distribution with df = \nu degrees of freedom has density

f(x) = \frac{\Gamma ((\nu+1)/2)}{\sqrt{\pi \nu} \Gamma (\nu/2)} (1 + x^2/\nu)^{-(\nu+1)/2}%

for all real x. It has mean 0 (for \nu > 1) and variance \frac{\nu}{\nu-2} (for \nu > 2). C.f. rt

Objects from the Class

Objects can be created by calls of the form Td(df). This object is a t distribution.

Slots

img

Object of class "Reals": The domain of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "TParameter": the parameter of this distribution (df), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rt)

d

Object of class "function": density function (calls function dt)

p

Object of class "function": cumulative function (calls function pt)

q

Object of class "function": inverse of the cumulative function (calls function qt)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Methods

initialize

signature(.Object = "Td"): initialize method

df

signature(object = "Td"): returns the slot df of the parameter of the distribution

df<-

signature(object = "Td"): modifies the slot df of the parameter of the distribution

ncp

signature(object = "Td"): returns the slot ncp of the parameter of the distribution

ncp<-

signature(object = "Td"): modifies the slot ncp of the parameter of the distribution

Ad hoc methods

For R Version <2.3.0 ad hoc methods are provided for slots q, r if ncp!=0; for R Version >=2.3.0 the methods from package stats are used.

Note

The general non-central t with parameters (\nu,\delta) = (df, ncp) is defined as a the distribution of T_{\nu}(\delta) := \frac{U + \delta}{\chi_{\nu}/\sqrt{\nu}} where U and \chi_{\nu} are independent random variables, U \sim {\cal N}(0,1), and \chi^2_\nu is chi-squared, see rchisq.

The most used applications are power calculations for t-tests:
Let T= \frac{\bar{X} - \mu_0}{S/\sqrt{n}} where \bar{X} is the mean and S the sample standard deviation (sd) of X_1,X_2,\dots,X_n which are i.i.d. N(\mu,\sigma^2). Then T is distributed as non-centrally t with df= n-1 degrees of freedom and non-centrality parameter ncp= (\mu - \mu_0) \sqrt{n}/\sigma.

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

TParameter-class, AbscontDistribution-class, Reals-class, rt

Examples

T <- Td(df = 1) # T is a t distribution with df = 1.
r(T)(1) # one random number generated from this distribution, e.g. -0.09697573
d(T)(1) # Density of this distribution is 0.1591549 for x = 1.
p(T)(1) # Probability that x < 1 is 0.75.
q(T)(.1) # Probability that x < -3.077684 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
df(T) # df of this distribution is 1.
df(T) <- 2 # df of this distribution is now 2.
Tn <- Td(df = 1, ncp = 5) 
  # T is a noncentral t distribution with df = 1 and ncp = 5.
d(Tn)(1) ## from R 2.3.0 on ncp no longer ignored...

Methods for function Truncate in Package ‘distr’

Description

Truncate-methods

Usage

Truncate(object, ...)
## S4 method for signature 'AbscontDistribution'
Truncate(object, lower = -Inf, upper = Inf)
## S4 method for signature 'DiscreteDistribution'
Truncate(object, lower= -Inf, upper = Inf)
## S4 method for signature 'LatticeDistribution'
Truncate(object, lower= -Inf, upper = Inf)
## S4 method for signature 'UnivarLebDecDistribution'
Truncate(object, lower = -Inf, upper = Inf, 
                    withSimplify = getdistrOption("simplifyD"))

Arguments

Value

the corresponding distribution of the truncated random variable

Methods

Truncate

signature(object = "AbscontDistribution"): returns the distribution of min(upper,max(X,lower)) conditioned to lower<=X<=upper, if X is distributed according to object; if slot .logExact of argument object is TRUE and if either there is only one-sided truncation or both truncation points lie on the same side of the median, we use this representation to enhance the range of applicability, in particular, for slot r, we profit from Peter Dalgaard's clever log-tricks as indicated in https://stat.ethz.ch/pipermail/r-help/2008-September/174321.html. To this end we use the internal functions (i.e.; non exported to namespace) .trunc.up and .trunc.low which provide functional slots r,d,p,q for one-sided truncation. In case of two sided truncation, we simply use one-sided truncation successively — first left and then right in case we are right of the median, and the other way round else; the result is again of class "AbscontDistribution";

Truncate

signature(object = "DiscreteDistribution"): returns the distribution of min(upper,max(X,lower)) conditioned to lower<=X<=upper, if X is distributed according to object; the result is again of class "DiscreteDistribution"

Truncate

signature(object = "LatticeDistribution"): if length of the corresp. lattice is infinite and slot .logExact of argument object is TRUE, we proceed similarly as in case of AbscontDistribution, also using internal functions .trunc.up and .trunc.low; else we use the corresponding "DiscreteDistribution" method; the result is again of class "LatticeDistribution"

Truncate

signature(object = "UnivarLebDecDistribution"): returns the distribution of min(upper,max(X,lower)) conditioned to lower<=X<=upper, if X is distributed according to object; the result is again of class "UnivarLebDecDistribution"

See Also

Huberize, Minimum

Examples

plot(Truncate(Norm(),lower=-1,upper=2))
TN <- Truncate(Norm(),lower=15,upper=15.7) ### remarkably right!
plot(TN)
r(TN)(30)
TNG <- Truncate(Geom(prob=0.05),lower=325,upper=329) ### remarkably right!
plot(TNG)

Class "UniNormParameter"

Description

The parameter of a univariate normal distribution, used by Norm-class

Objects from the Class

Objects can be created by calls of the form new("NormParameter", sd, mean). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Norm is instantiated.

Slots

sd

Object of class "numeric": the sd of a univariate normal distribution

mean

Object of class "numeric": the mean of a univariate normal distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "NormParameter", directly. Class "Parameter", by class "NormParameter".

Methods

initialize

signature(.Object = "UniNormParameter"): initialize method

mean

signature(object = "UniNormParameter"): returns the slot mean of the parameter of the distribution

mean<-

signature(object = "UniNormParameter"): modifies the slot mean of the parameter of the distribution

sd

signature(object = "UniNormParameter"): returns the slot sd of the parameter of the distribution

sd<-

signature(object = "UniNormParameter"): modifies the slot sd of the parameter of the distribution

Author(s)

Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Matthias Kohl Matthias.Kohl@stamats.de

See Also

Norm-class NormParameter-class Parameter-class

Examples

W <- new("UniNormParameter", mean = 0, sd = 1)
sd(W) # sd of this distribution is 1
sd(W) <- 2 # sd of this distribution is now 2

Class "Unif"

Description

The uniform distribution has density

d(x) = \frac{1}{max-min}

for min, by default =0, \le x \le max, by default =1. C.f. runif

Objects from the Class

Objects can be created by calls of the form Unif(Min, Max). This object is a uniform distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "UnifParameter": the parameter of this distribution (Min and Max), declared at its instantiation

r

Object of class "function": generates random numbers (calls function runif)

d

Object of class "function": density function (calls function dunif)

p

Object of class "function": cumulative function (calls function punif)

q

Object of class "function": inverse of the cumulative function (calls function qunif)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Is-Relations

By means of setIs, R “knows” that a distribution object obj of class "Unif" with Min 0 and Max 1 also is a Beta distribution with parameters shape1 = 1, shape2 = 1, ncp = 0.

Methods

initialize

signature(.Object = "Unif"): initialize method

Min

signature(object = "Unif"): returns the slot Min of the parameter of the distribution

Min<-

signature(object = "Unif"): modifies the slot Min of the parameter of the distribution

Max

signature(object = "Unif"): returns the slot Max of the parameter of the distribution

Max<-

signature(object = "Unif"): modifies the slot Max of the parameter of the distribution

*

signature(e1 = "Unif", e2 = "numeric"): multiplication of this uniform distribution by an object of class ‘numeric’

+

signature(e1 = "Unif", e2 = "numeric"): addition of this uniform distribution to an object of class ‘numeric’