Description Usage Arguments Details Value Author(s) References Examples

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.

1 2 3 4 5 6 7 |

```
Loading required package: quantreg
Loading required package: SparseM
Attaching package: 'SparseM'
The following object is masked from 'package:base':
backsolve
$`-2LLR`
[1] 1.4959
$Pval
[1] 0.4733359
$lambda
[1] 0.16370747 0.05954525
$grad
[1] 3.814455e-08 -1.031469e-07
$hess
[,1] [,2]
[1,] 57.590072 -9.411123
[2,] -9.411123 54.854378
$wts
[1] 1.0938250 1.2738451 1.0669185 0.9989470 0.8324758 1.4368976 1.0551255
[8] 1.1775916 0.9117519 0.8813374 0.8280284 1.0179621 1.0461412 0.7790994
[15] 1.0361341 1.0143722 0.8889165 0.8249367 0.7827030 1.0806904 0.9677678
[22] 1.1207849 1.4732085 1.1967321 1.1280005 1.0826161 0.7310015 0.8544029
[29] 0.9920329 0.8845332 1.1816435 0.9656110 0.8414424 0.7905337 1.0025622
[36] 0.9188318 0.9479182 0.7417220 1.2250821 0.9036137 0.8228518 0.7557114
[43] 1.3213910 0.9372755 1.2145213 1.1710812 1.1887490 0.8907115 0.9034953
[50] 0.8164714
$nits
[1] 4
$`-2LLR`
[1] 0.7072538
$Pval
[1] 0.400357
$lambda
[1] 0.04691874
$grad
[1] -1.294647e-15
$hess
[,1]
[1,] 345.9248
$wts
[1] 1.2630322 1.0986891 0.9748913 0.9726342 0.9957298 1.2072389 0.9961728
[8] 1.1188140 0.8840956 0.8866232 0.9400486 1.0657325 1.0455518 0.9844367
[15] 0.9495881 1.0109925 0.9163291 0.9021504 0.9801274 1.0506449 0.8724263
[22] 1.0147955 1.2591159 1.0868785 1.1261789 1.4632319 0.9203709 0.8064722
[29] 0.9323382 0.9278363 1.0529094 0.9620706 0.8645996 0.9186668 0.9859657
[36] 0.8618430 0.8024267 0.8839918 1.0825990 0.8802598 0.9121091 0.9283928
[43] 1.0945092 1.1226222 1.0026974 0.9993862 1.1051442 0.9690086 1.0475848
[50] 0.8700456
$nits
[1] 5
$`-2LLR`
[1] 1.805702
$Pval
[1] 0.1790247
$lambda
[1] 0.01078493
$grad
[1] 3.256062e-08
$hess
[,1]
[1,] 28962.89
$wts
[1] 0.8770359 2.7496028 0.8688180 0.8607526 0.8939472 0.8854108 0.9298048
[8] 0.9205734 0.8374306 0.9893301 0.9115235 0.8688180 0.9115235 0.7876352
[15] 0.9392232 0.8528356 0.8299349 1.2245056 0.9026498 0.9586443
$nits
[1] 7
```

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