# 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

1 | ```
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 |

`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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
## 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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