Empirical likelihood ratio confidence intervals for a single functional. (English) Zbl 0641.62032

Let \((X_ 1,...,X_ n)\) be a random sample, its components \(X_ i\) are observations from a distribution-function \(F_ 0\). The empirical distribution function \(F_ n\) is a nonparametric maximum likelihood estimate of \(F_ 0\). \(F_ n\) maximizes \[ L(F)=\prod^{n}_{i=1}\{F(X_ i)-F(X_ i-)\} \] over all distribution functions F. Let \(R(F)=L(F)/L(F_ n)\) be the empirical likelihood ratio function and T(.) any functional. It is shown that sets of the form \[ \{T(F)| R(F)\geq c\} \] may be used as confidence regions for some \(T(F_ 0)\) like the sample mean or a class of M-estimators (especially the quantiles of \(F_ 0)\). 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


62G15 Nonparametric tolerance and confidence regions
62G30 Order statistics; empirical distribution functions
62G05 Nonparametric estimation
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