| retel {retel} | R Documentation |
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 |
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 |
opts |
An optional list with optimization options for |
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)