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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

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