el.test {emplik} R Documentation

## Empirical likelihood ratio test for the means, uncensored data

### Description

Compute the empirical likelihood ratio with the mean vector fixed at mu.

The log empirical likelihood been maximized is

 \sum_{i=1}^n \log \Delta F(x_i).

### Usage

el.test(x, mu, lam, maxit=25, gradtol=1e-7,
svdtol = 1e-9, itertrace=FALSE)


### Arguments

 x a matrix or vector containing the data, one row per observation. mu a numeric vector (of length  = ncol(x)) to be tested as the mean vector of x above, as H_0. lam an optional vector of length  = length(mu), the starting value of Lagrange multipliers, will use 0 if missing. maxit an optional integer to control iteration when solve constrained maximization. gradtol an optional real value for convergence test. svdtol an optional real value to detect singularity while solve equations. itertrace a logical value. If the iteration history needs to be printed out.

### Details

If mu is in the interior of the convex hull of the observations x, then wts should sum to n. If mu is outside the convex hull then wts should sum to nearly zero, and -2LLR will be a large positive number. It should be infinity, but for inferential purposes a very large number is essentially equivalent. If mu is on the boundary of the convex hull then wts should sum to nearly k where k is the number of observations within that face of the convex hull which contains mu.

When mu is interior to the convex hull, it is typical for the algorithm to converge quadratically to the solution, perhaps after a few iterations of searching to get near the solution. When mu is outside or near the boundary of the convex hull, then the solution involves a lambda of infinite norm. The algorithm tends to nearly double lambda at each iteration and the gradient size then decreases roughly by half at each iteration.

The goal in writing the algorithm was to have it “fail gracefully" when mu is not inside the convex hull. The user can either leave -2LLR “large and positive" or can replace it by infinity when the weights do not sum to nearly n.

### Value

A list with the following components:

 -2LLR the -2 loglikelihood ratio; approximate chisq distribution under H_o. Pval the observed P-value by chi-square approximation. lambda the final value of Lagrange multiplier. grad the gradient at the maximum. hess the Hessian matrix. wts weights on the observations nits number of iteration performed

### Author(s)

Original Splus code by Art Owen. Adapted to R by Mai Zhou.

### References

Owen, A. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18, 90-120.

### Examples

x <- matrix(c(rnorm(50,mean=1), rnorm(50,mean=2)), ncol=2,nrow=50)
el.test(x, mu=c(1,2))
## Suppose now we wish to test Ho: 2mu(1)-mu(2)=0, then
y <- 2*x[,1]-x[,2]
el.test(y, mu=0)
xx <- c(28,-44,29,30,26,27,22,23,33,16,24,29,24,40,21,31,34,-2,25,19)
el.test(xx, mu=15)  #### -2LLR = 1.805702


[Package emplik version 1.1-1 Index]