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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

Cited in: Zbl 1131.62106 Zbl 1110.62062 Zbl 1080.62022 Zbl 1073.62044 Zbl 1091.62038 Zbl 1047.62026 Zbl 1009.62092 Zbl 0970.62030 Zbl 1054.62550 Zbl 0943.62100 Zbl 1076.62511 Zbl 0893.62041 Zbl 0899.62061 Zbl 0824.62029 Zbl 0802.62052 Zbl 0791.62052 Zbl 0753.62026

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Zentralblatt MATH Berlin [Germany]