Uniform convergence of the empirical cumulative distribution function under informative selection from a finite population

TL;DR

This paper establishes conditions under which the empirical cumulative distribution function converges uniformly to a weighted superpopulation c.d.f. in the context of informative sampling from finite populations, extending classical results.

Contribution

It provides a theoretical framework and verifiable conditions for uniform convergence of the empirical c.d.f. under informative selection mechanisms, generalizing the Glivenko-Cantelli theorem.

Findings

Uniform convergence in $L_2$ and almost surely

Conditions applicable to real survey designs

Extension of Glivenko-Cantelli theorem to informative sampling

Abstract

Consider informative selection of a sample from a finite population. Responses are realized as independent and identically distributed (i.i.d.) random variables with a probability density function (p.d.f.) f, referred to as the superpopulation model. The selection is informative in the sense that the sample responses, given that they were selected, are not i.i.d. f. In general, the informative selection mechanism may induce dependence among the selected observations. The impact of such dependence on the empirical cumulative distribution function (c.d.f.) is studied. An asymptotic framework and weak conditions on the informative selection mechanism are developed under which the (unweighted) empirical c.d.f. converges uniformly, in $L_2$ and almost surely, to a weighted version of the superpopulation c.d.f. This yields an analogue of the Glivenko-Cantelli theorem. A series of examples, motivated by real problems in surveys and other observational studies, shows that the conditions are verifiable for specified designs.