This program is similar to `el.test( )`

except it takes weights.

The mean constraints are:

* ∑_{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. *

1 2 | ```
el.test.wt2(x, wt, mu, maxit = 25, gradtol = 1e-07, Hessian = FALSE,
svdtol = 1e-09, itertrace = FALSE)
``` |

`x` |
a matrix (of size nxp) or vector containing the observations. |

`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. |

`mu` |
a vector of length p, used in the constraint. weighted
mean value of |

`maxit` |
an integer, the maximum number of iteration. |

`gradtol` |
a positive real number, the tolerance for a solution |

`Hessian` |
logical. if the Hessian needs to be computed? |

`svdtol` |
tolerance in perform SVD of the Hessian matrix. |

`itertrace` |
TRUE/FALSE, if the intermediate steps needs to be printed. |

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

It is similar to the function `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).

A list with the following components:

`lambda` |
the Lagrange multiplier. Solution. |

`wt` |
the vector of weights. |

`grad` |
The gradian at the final solution. |

`nits` |
number of iterations performed. |

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

Mai Zhou

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*

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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