Type: Package
Title: Regularized Exponentially Tilted Empirical Likelihood
Version: 0.1.1
Description: Implements the regularized exponentially tilted empirical likelihood method. Details of the method are given in Kim, MacEachern, and Peruggia (2023) <doi:10.48550/arXiv.2312.17015>. This work was supported by the U.S. National Science Foundation under Grants No. SES-1921523 and DMS-2015552.
License: GPL (≥ 3)
URL: https://github.com/markean/retel
BugReports: https://github.com/markean/retel/issues
Depends: R (≥ 4.1.0)
Suggests: melt, spelling
Encoding: UTF-8
Language: en-US
LazyData: true
RoxygenNote: 7.3.2
Imports: checkmate, Matrix, matrixcalc, nloptr
NeedsCompilation: no
Packaged: 2025-07-27 15:12:12 UTC; markean
Author: Eunseop Kim ORCID iD [aut, cph, cre], Steven MacEachern [ctb, ths], Mario Peruggia [ctb, ths]
Maintainer: Eunseop Kim <markean@pm.me>
Repository: CRAN
Date/Publication: 2025-07-27 15:30:02 UTC

retel: Regularized Exponentially Tilted Empirical Likelihood

Description

Implements the regularized exponentially tilted empirical likelihood method. Details of the method are given in Kim, MacEachern, and Peruggia (2023) doi:10.48550/arXiv.2312.17015. This work was supported by the U.S. National Science Foundation under Grants No. SES-1921523 and DMS-2015552.

Author(s)

Maintainer: Eunseop Kim markean@pm.me (ORCID) [copyright holder]

Other contributors:

References

Kim E, MacEachern SN, Peruggia M (2023). "Regularized Exponentially Tilted Empirical Likelihood for Bayesian Inference." doi:10.48550/arXiv.2312.17015.

See Also

Useful links:

Exponentially tilted empirical likelihood

Description

Computes exponentially tilted empirical likelihood.

Usage

etel(fn, x, par, opts = NULL)

Arguments

fn

An estimating function that takes the data x and parameter value par as its arguments, returning a numeric matrix. Each row is the return value from the corresponding row in x.

x

A numeric matrix, or an object that can be coerced to a numeric matrix. Each row corresponds to an observation. The number of rows must be greater than the number of columns.

par

A numeric vector of parameter values to be tested.

opts

An optional list with optimization options for nloptr::nloptr(). Defaults to NULL.

Value

A single numeric value representing the log-likelihood ratio. It contains the optimization results as the attribute optim.

References

Schennach, SM (2005). "Bayesian Exponentially Tilted Empirical Likelihood." Biometrika, 92, 31–46.

Examples

# Generate data
set.seed(63456)
x <- rnorm(100)

# Define an estimating function (ex. mean)
fn <- function(x, par) {
  x - par
}

# Set parameter value
par <- 0

# Call the etel function
etel(fn, x, par)

Median Income for 4-Person Families in the USA

Description

A dataset of median income for 4-person families by state.

Usage

data("income")

Format

A data frame with 51 rows and 6 columns:

state

States, including the District of Columbia.

mi_1979

Estimated median income for 4-person families in 1979 (standardized).

mi_1989

Estimated median income for 4-person families in 1989 (standardized).

pci_1979

Per capita income in 1979.

pci_1989

Per capita income in 1989.

ami

Census median income in 1979, adjusted for per capita income (standardized).

Source

https://www.census.gov/data/tables/time-series/demo/income-poverty/4-person.html

Examples

data("income")
income

Regularized exponentially tilted empirical likelihood

Description

Computes regularized exponentially tilted empirical likelihood.

Usage

retel(fn, x, par, mu, Sigma, tau, type = "full", opts = NULL)

Arguments

fn

An estimating function that takes the data x and parameter value par as its arguments, returning a numeric matrix. Each row is the return value from the corresponding row in x.

x

A numeric matrix, or an object that can be coerced to a numeric matrix. Each row corresponds to an observation. The number of rows must be greater than the number of columns.

par

A numeric vector of parameter values to be tested.

mu

A numeric vector of parameters for regularization. See 'Details' for more information.

Sigma

A numeric matrix, or an object that can be coerced to a numeric matrix, of parameters for regularization. See 'Details' for more information.

tau

A single numeric parameter for regularization. See 'Details' for more information.

type

A single character indicating the type of regularized exponentially tilted empirical likelihood. It must be either "full" or "reduced". Defaults to "full". See 'Details' for more information.

opts

An optional list with optimization options for nloptr::nloptr(). Defaults to NULL.

Details

Let \{\bm{X}_i\}_{i = 1}^n denote independent d_x-dimensional observations from a complete probability space {(\mathcal{X}, \mathcal{F}, P)} satisfying the moment condition:

\textnormal{E}_P[\bm{g}(\bm{X}_i, \bm{\theta})] = \bm{0},

where {\bm{g}}: {\mathbb{R}^{d_x} \times \Theta} \mapsto {\mathbb{R}^p} is an estimating function with the true parameter value {\bm{\theta}_0} \in {\Theta} \subset \mathbb{R}^p.

For a given parameter value \bm{\theta}, regularized exponentially tilted empirical likelihood solves the following optimization problem:

\min_{\bm{\lambda} \in \mathbb{R}^p} \left\{ d_n\left(\bm{\theta}, \bm{\lambda}\right) + p_n\left(\bm{\theta}, \bm{\lambda}\right) \right\},

where

d_n\left(\bm{\theta}, \bm{\lambda}\right) = \frac{1}{n + \tau_n} \sum_{i = 1}^n \exp \left( \bm{\lambda}^\top \bm{g}\left(\bm{X}_i, \bm{\theta}\right) \right)

and

p_n\left(\bm{\theta}, \bm{\lambda}\right) = \frac{\tau_n}{n + \tau_n} \exp \left( \bm{\lambda}^\top\bm{\mu}_{n, \bm{\theta}} + \frac{1}{2} \bm{\lambda}^\top\bm{\Sigma}_{n, \bm{\theta}}\bm{\lambda} \right).

Here, {\tau_n} > {0}, \bm{\mu}_{n, \bm{\theta}}, \bm{\Sigma}_{n, \bm{\theta}} are all tuning parameters that control the strength of {p_n(\bm{\theta}, \bm{\lambda})} as a penalty.

Once we have determined the solution {\bm{\lambda}_{RET}}, we define the likelihood ratio function as follows:

R_{RET}\left(\bm{\theta}\right) = \left( \frac{n + \tau_n}{\tau_n}p_c\left(\bm{\theta}\right)\right) \prod_{i = 1}^n \left(n + \tau_n\right)p_i\left(\bm{\theta} \right),

where

p_i\left(\bm{\theta}\right) = \frac{\exp \left( {\bm{\lambda}_{RET}}^\top\bm{g}\left(\bm{X}_i, \bm{\theta}\right) \right) }{c_n\left(\bm{\theta}, \bm{\lambda}_{RET}\right) } \quad \left(i = 1, \dots, n\right),\quad p_c\left(\bm{\theta}\right) = \frac{p_n \left(\bm{\theta}, \bm{\lambda}_{RET}\right) }{c_n\left(\bm{\theta}, \bm{\lambda}_{RET}\right) },

and c_n\left(\bm{\theta}, \bm{\lambda}_{RET}\right) = d_n\left(\bm{\theta}, \bm{\lambda}_{RET}\right) + p_n\left(\bm{\theta}, \bm{\lambda}_{RET}\right) . The reduced version of the likelihood ratio function is defined as:

\widetilde{R}_{RET}\left(\bm{\theta}\right) = \prod_{i = 1}^n \left(n + \tau_n\right)p_i\left(\bm{\theta}\right).

See the references below for more details on derivation, interpretation, and properties.

Value

A single numeric value representing the log-likelihood ratio. It contains the optimization results as the attribute optim.

References

Kim E, MacEachern SN, Peruggia M (2023). "Regularized Exponentially Tilted Empirical Likelihood for Bayesian Inference." doi:10.48550/arXiv.2312.17015.

Examples

# Generate data
set.seed(63456)
x <- rnorm(100)

# Define an estimating function (ex. mean)
fn <- function(x, par) {
  x - par
}

# Set parameter value
par <- 0

# Set regularization parameters
mu <- 0
Sigma <- 1
tau <- 1

# Call the retel function
retel(fn, x, par, mu, Sigma, tau)