retel: Regularized Exponentially Tilted Empirical Likelihood

Description

Implements the regularized exponentially tilted empirical likelihood method. Details of the methods 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 [copyright holder]

Other contributors:

• Steven MacEachern [contributor, thesis advisor]

• Mario Peruggia [contributor, thesis advisor]

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:

• https://github.com/markean/retel

• Report bugs at https://github.com/markean/retel/issues

Exponentially tilted empirical likelihood

Description

Computes exponentially tilted empirical likelihood.

Usage

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

Arguments

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

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)