Type: Package
Title: The Discrete Stable Distribution Functions
Version: 0.1.0
Maintainer: Krutto Annika <annikakrutto@gmail.com>
Description: Probability generating function, formulae for the probabilities (discrete density) and random generation for discrete stable random variables.
License: GPL-3
Encoding: UTF-8
Imports: stabledist, stats, Rdpack
RdMacros: Rdpack
RoxygenNote: 7.2.3
NeedsCompilation: no
Packaged: 2023-05-02 19:57:13 UTC; annikakrutto
Author: Krutto Annika ORCID iD [aut, cre]
Repository: CRAN
Date/Publication: 2023-05-03 18:50:05 UTC

dstabledist: The Discrete Stable Distribution Functions

Description

Probability generating function, formulae for the probabilities (discrete density) and random generation for discrete stable random variables.

Author(s)

Maintainer: Krutto Annika annikakrutto@gmail.com (ORCID)

The discrete stable distribution: formulae for the probabilities (density)

Description

Computes the value of the formulae for the probabilities (density) of a discrete stable distribution DS(alpha,lambda), by combining the explicit and fast asymptotic formulae.

Usage

ddstable(x, alpha, lambda)

Arguments

x

a vector of non-negative integer quantiles, k>=0

alpha

tail index parameter alpha in the interval= (0, 1]

lambda

positive location parameter lambda>0

Value

Returns the value of the formulae for the probabilities (density) of DS(alpha,lambda).

References

Christoph G, Schreiber K (1998). “Discrete stable random variables.” Statistics & Probability Letters, 37(3), 243-247. ISSN 0167-7152, doi:10.1016/S0167-7152(97)00123-5.

Examples

ddstable(c(0,1,2,100),1,lambda=1)#This is Poisson with lambda=1
dpois(c(0,1,2,100),1)#Checking with dpois
ddstable(c(0,1,2,100),0.5,lambda=1) # tail is heavier
ddstable(c(0,1,2,3,6,100),0.5,lambda=3) # change in location

The discrete stable distribution: probability generating function

Description

Computes probability generating function of a discrete stable distribution DS(alpha,lambda).

Usage

pgdstable(z, alpha, lambda = 1)

Arguments

z

argument of probability generating function, z in the interval= [-1, 1].

alpha

tail index parameter alpha in the interval= (0, 1].

lambda

positive location parameter lambda>0.

Value

Returns value of probability generating function of DS(alpha,lambda). A warning is displayed for invalid parameter values.

References

Steutel FW, van Harn K (1979). “Discrete Analogues of Self-Decomposability and Stability.” The Annals of Probability, 7(5), 893 – 899. doi:10.1214/aop/1176994950.

Examples

pgdstable(c(-1,0,1),0.5,1)
pgdstable(c(-1,0,1),1,1) #This is Poisson
 curve(pgdstable(x,1,lambda=1), c(-1,1),col=1,ylab='prob. gen. fun.',xlab='z')
 curve(pgdstable(x,0.5,lambda=1), c(-1,1),col=2,add=TRUE)
 curve(pgdstable(x,0.2,lambda=1), c(-1,1),col=4,add=TRUE)
legend('topleft',legend=c(1,0.5,0.1), col=c(1,2,4), lty = 1, title='alpha')

The discrete stable distribution: random generation

Description

Generates random variates from a discrete stable distribution DS(alpha,lambda).

Usage

rdstable(n, alpha, lambda = 1)

Arguments

n

number of random values to return.

alpha

tail index parameter alpha in the interval= (0, 1]

lambda

positive location parameter lambda>0

Value

returns random variates from DS(alpha,lambda). A warning is displayed for invalid parameter values.

References

Devroye L (1993). “A triptych of discrete distributions related to the stable law.” Statistics & Probability Letters, 18(5), 349-351. ISSN 0167-7152, doi:10.1016/0167-7152(93)90027-G.

Examples

rdstable(10,alpha=1,lambda=1) #this is Poisson
rdstable(10,alpha=0.5,lambda=1) # heavier tail more prone to extremes
rdstable(10,alpha=0.1,lambda=1) # heavier tail more prone to extremes