\name{el.test.wt2}
\alias{el.test.wt2}
\title{Weighted Empirical Likelihood ratio for mean(s), uncensored data}
\usage{
el.test.wt2(x, wt, mu, maxit = 25, gradtol = 1e-07, Hessian = FALSE, 
    svdtol = 1e-09, itertrace = FALSE)
}
\description{
This program is similar to \code{el.test( )} except it takes weights.

The mean constraints are:
\deqn{ \sum_{i=1}^n  p_i x_i  = \mu . }
Where \eqn{p_i = \Delta F(x_i)} is a probability.
Plus the probability constraint: \eqn{ \sum p_i =1}.

The weighted log empirical likelihood been maximized is
\deqn{ \sum_{i=1}^n w_i \log p_i. } 
}
\arguments{
    \item{x}{a matrix (of size nxp) or vector containing the observations.}
    \item{wt}{a vector of length n, containing the weights. If weights are 
              all 1, this is very simila to el.test. wt have to be positive.}
    \item{mu}{a vector of length p, used in the constraint. weighted
                mean value of \eqn{f(X)}.}
    \item{maxit}{an integer, the maximum number of iteration.}
    \item{gradtol}{a positive real number, the tolerance for a solution}
    \item{Hessian}{logical. if the Hessian needs to be computed?}
    \item{svdtol}{ tolerance in perform SVD of the Hessian matrix. }
    \item{itertrace}{TRUE/FALSE, if the intermediate steps needs to be printed.}
}
\value{
    A list with the following components:
    \item{lambda}{the Lagrange multiplier. Solution.}
    \item{wt}{the vector of weights.}
    \item{grad}{The gradian at the final solution.}
    \item{nits}{number of iterations performed. }
    \item{prob}{The probabilities that maximized the weighted 
                  empirical likelihood under mean constraint.}
}
\details{
This function used to be an internal function. 
It becomes external because others may find it useful.

It is similar to the function \code{el.test( )} with the
 following differences: 

(1) The output lambda in el.test.wts, when divided by n
(the sample size or sum of all the weights) should be equal to the
output lambda in el.test.

(2) The Newton step of iteration in el.test.wts is different from
those in el.test. (even when all the weights are one).

}
\author{ Mai Zhou }
\references{
Owen, A. (1990). Empirical likelihood ratio confidence regions. 
    \emph{Ann. Statist.} \bold{18}, 90-120.

Zhou, M. (2005). Empirical likelihood ratio with arbitrary censored/truncated data by EM algorithm. 
 \emph{Journal of Computational and Graphical Statistics}, 
\bold{14}, 643-656. 

   Zhou, M. (2002). 
        Computing censored empirical likelihood ratio 
        by EM algorithm. 
    \emph{Tech Report, Univ. of Kentucky, Dept of Statistics}
}
\examples{
## 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)
}
\keyword{nonparametric}
\keyword{htest}