Weighted Empirical Likelihood ratio for mean, uncensored data

Description

This program is similar to el.test( ) except it takes weights, and is for one dimensional mu.

The mean constraint considered is:

∑_{i=1}^n p_i x_i = μ .

where p_i = Δ F(x_i) is a probability. Plus the probability constraint: ∑ p_i =1.

The weighted log empirical likelihood been maximized is

∑_{i=1}^n w_i \log p_i.

Usage

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el.test.wt(x, wt, mu, usingC=TRUE)

Arguments

x

a vector containing the observations.

wt

a vector containing the weights.

mu

a real number used in the constraint, weighted mean value of f(X).

usingC

TRUE: use C function, which may be benifit when sample size is large; FALSE: use pure R function.

Details

This function used to be an internal function. It becomes external because others may find it useful elsewhere.

The constant mu must be inside ( \min x_i , \max x_i ) for the computation to continue.

Value

A list with the following components:

x

the observations.

wt

the vector of weights.

prob

The probabilities that maximized the weighted empirical likelihood under mean constraint.

Author(s)

Mai Zhou, Y.F. Yang for C part.

References

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

Zhou, M. (2002). Computing censored empirical likelihood ratio by EM algorithm. Tech Report, Univ. of Kentucky, Dept of Statistics

Examples

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## example with tied observations
x <- c(1, 1.5, 2, 3, 4, 5, 6, 5, 4, 1, 2, 4.5)
d <- c(1,   1, 0, 1, 0, 1, 1, 1, 1, 0, 0,   1)
el.cen.EM(x,d,mu=3.5)
## we should get "-2LLR" = 1.2466....
myfun5 <- function(x, theta, eps) {
u <- (x-theta)*sqrt(5)/eps 
INDE <- (u < sqrt(5)) & (u > -sqrt(5)) 
u[u >= sqrt(5)] <- 0 
u[u <= -sqrt(5)] <- 1 
y <- 0.5 - (u - (u)^3/15)*3/(4*sqrt(5)) 
u[ INDE ] <- y[ INDE ] 
return(u)
}
el.cen.EM(x, d, fun=myfun5, mu=0.5, theta=3.5, eps=0.1)

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