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).
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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 H_o. |
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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