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Empirical likelihood ratio confidence intervals for a single functional. (English) Zbl 0641.62032
Let $(X\sb 1,...,X\sb n)$ be a random sample, its components $X\sb i$ are observations from a distribution-function $F\sb 0$. The empirical distribution function $F\sb n$ is a nonparametric maximum likelihood estimate of $F\sb 0$. $F\sb n$ maximizes $$ L(F)=\prod\sp{n}\sb{i=1}\{F(X\sb i)-F(X\sb i-)\} $$ over all distribution functions F. Let $R(F)=L(F)/L(F\sb n)$ be the empirical likelihood ratio function and T(.) any functional. It is shown that sets of the form $$ \{T(F)\vert R(F)\ge c\} $$ may be used as confidence regions for some $T(F\sb 0)$ like the sample mean or a class of M-estimators (especially the quantiles of $F\sb 0)$. These confidence intervals are compared in a simulation study to some bootstrap confidence intervals and to confidence intervals based on a t-statistic for a confidence coefficient $1-\alpha =0.9$. It seems that two of the bootstrap intervals may be recommended but the simulation is based on 1000 runs only.
Reviewer: D. Rasch

62G15Nonparametric tolerance and confidence regions
62G30Order statistics; empirical distribution functions
62G05Nonparametric estimation
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