compute_AEL {VBel} | R Documentation |
Compute the Adjusted Empirical Likelihood
Description
Evaluates the Log-Adjusted Empirical Likelihood (AEL) (Chen, Variyath, and Abraham 2008) for a given data set, moment conditions and parameter values. The AEL function is formulated as
\log \text{AEL}(\boldsymbol{\theta}) = \max_{\mathbf{w}'} \sum\limits_{i=1}^{n+1} \log(w_i'),
where \mathbf{z}_{n+1}
is a pseudo-observation that satisfies
h(\mathbf{z}_{n+1}, \boldsymbol{\theta}) = -\frac{a_n}{n} \sum\limits_{i=1}^n h(\mathbf{z}_i, \boldsymbol{\theta})
for some constant a_n > 0
that may (but not necessarily) depend on n
, and \mathbf{w}' = (w_1', \ldots, w_n', w_{n+1}')
is a vector of probability weights that define a discrete distribution on \{\mathbf{z}_1, \ldots, \mathbf{z}_n, \mathbf{z}_{n+1}\}
, and are subject to the constraints
\sum\limits_{i=1}^{n+1} w_i' h(\mathbf{z}_i, \boldsymbol{\theta}) = 0, \quad \text{and} \quad \sum\limits_{i=1}^{n+1} w_i' = 1.
Here, the maximizer \tilde{\mathbf{w}}
is of the form
\tilde{w}_i = \frac{1}{n+1} \frac{1}{1 + \lambda_{\text{AEL}}^\top h(\mathbf{z}_i, \boldsymbol{\theta})},
where \lambda_{\text{AEL}}
satisfies the constraints
\frac{1}{n+1} \sum\limits_{i=1}^{n+1} \frac{h(\mathbf{z}_i, \boldsymbol{\theta})}{1 + \lambda_{\text{AEL}}^\top h(\mathbf{z}_i, \boldsymbol{\theta})} = 0, \quad \text{and} \quad
\frac{1}{n+1} \sum\limits_{i=1}^{n+1} \frac{1}{1 + \lambda_{\text{AEL}}^\top h(\mathbf{z}_i, \boldsymbol{\theta})} = 1.
Usage
compute_AEL(th, h, lam0, a, z, iters = 500, returnH = FALSE)
Arguments
th |
p x 1 parameter vector to evaluate the AEL function at |
h |
User-defined moment-condition function. Note that output should be an (n-1) x K matrix where K is necessarily |
lam0 |
Initial vector for Lagrange multiplier lambda |
a |
Positive scalar adjustment constant |
z |
(n-1) x d data matrix. Note that |
iters |
Number of iterations using Newton-Raphson for estimation of lambda. Default: |
returnH |
Whether to return calculated values of h, H matrix and lambda. Default: 'FALSE |
Details
Note that theta (th
) is a p-dimensional vector, h
is a K-dimensional vector and K \geq
p
Value
A numeric value for the Adjusted Empirical Likelihood function computed evaluated at a given theta value
Author(s)
Weichang Yu, Jeremy Lim
References
Chen, J., Variyath, A. M., and Abraham, B. (2008), “Adjusted Empirical Likelihood and its Properties,” Journal of Computational and Graphical Statistics, 17, 426–443. Pages 2,3,4,5,6,7 doi:10.1198/106186008X321068
Examples
# Generating 30 data points from a simple linear-regression model
set.seed(1)
x <- runif(30, min = -5, max = 5)
vari <- rnorm(30, mean = 0, sd = 1)
y <- 0.75 - x + vari
z <- cbind(x, y)
lam0 <- matrix(c(0,0), nrow = 2)
th <- matrix(c(0.8277, -1.0050), nrow = 2)
# Specify AEL constant and Newton-Rhapson iteration
a <- 0.00001
iters <- 10
# Specify moment condition functions for linear regression
h <- function(z, th) {
xi <- z[1]
yi <- z[2]
h_zith <- c(yi - th[1] - th[2] * xi, xi*(yi - th[1] - th[2] * xi))
matrix(h_zith, nrow = 2)
}
result <- compute_AEL(th, h, lam0, a, z, iters)