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

Vardi, Y.

Ann. Stat. 13, 178-205 (1985).

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

Keywords:

empirical distributions; sample selection bias; weighted; distributions; nonparametric maximum likelihood estimation; sufficiency