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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)= i=1 n {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)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-α=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:
62G15Nonparametric tolerance and confidence regions
62G30Order statistics; empirical distribution functions
62G05Nonparametric estimation