Owen, Art B. Empirical likelihood ratio confidence intervals for a single functional. (English) Zbl 0641.62032 Biometrika 75, No. 2, 237-249 (1988). 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 Cited in 27 ReviewsCited in 999 Documents MSC: 62G15 Nonparametric tolerance and confidence regions 62G30 Order statistics; empirical distribution functions 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 PDFBibTeX XMLCite \textit{A. B. Owen}, Biometrika 75, No. 2, 237--249 (1988; Zbl 0641.62032) Full Text: DOI