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Empirical distributions in selection bias models. (English) Zbl 0578.62047

Assume that in each of s samples, random variables \(y_{i1},...,y_{in_ i}\), \(1\leq i\leq s\), are generated from some unknown distribution function (d.f.) F, according to known sampling rules \(w_ i\), \(1\leq i\leq s\), i.e. in the i-th sample, the y’s have \(F_ i(t)=W_ i^{-1}(F)\int^{t}_{-\infty}w_ i(u)dF(u)\) as their common d.f. Here \(W_ i(F)=\int^{\infty}_{-\infty}w_ i(u)dF(u)\) serves as a norming constant.
The author considers nonparametric maximum likelihood estimation of F, and derives a necessary and sufficient condition for the existence and uniqueness of the estimator \(\hat F.\) Also sufficiency of \(\hat F\) is proved.
Reviewer: W.Stute

MSC:

62G05 Nonparametric estimation
62E99 Statistical distribution theory
62G30 Order statistics; empirical distribution functions
62M99 Inference from stochastic processes
62P10 Applications of statistics to biology and medical sciences; meta analysis
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