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

∑_{i=1}^n \log Δ F(x_i).

Usage

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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

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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

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