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Zbl 0641.62032
Owen, Art B.
Empirical likelihood ratio confidence intervals for a single functional.
(English)
[J] Biometrika 75, No.2, 237-249 (1988). ISSN 0006-3444; ISSN 1464-3510/e

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.
[D.Rasch]
MSC 2000:
*62G15 Nonparametric confidence regions, etc.
62G30 Order statistics, etc.
62G05 Nonparametric estimation

Keywords: differentiable statistical functionals; empirical distribution; nonparametric maximum likelihood estimate; empirical likelihood ratio function; confidence regions; sample mean; M-estimators; quantiles; bootstrap confidence intervals; t-statistic

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