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).*

1 2 |

`x` |
a matrix or vector containing the data, one row per observation. |

`mu` |
a numeric vector (of length |

`lam` |
an optional vector of length |

`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. |

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.

A list with the following components:

`-2LLR` |
the -2 loglikelihood ratio; approximate chisq distribution
under |

`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 |

Original Splus code by Art Owen. Adapted to R by Mai Zhou.

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

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