Open Access
Translator Disclaimer
November 2012 Uniform convergence of the empirical cumulative distribution function under informative selection from a finite population
Daniel Bonnéry, F. Jay Breidt, François Coquet
Bernoulli 18(4): 1361-1385 (November 2012). DOI: 10.3150/11-BEJ369

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.

Citation

Download Citation

Daniel Bonnéry. F. Jay Breidt. François Coquet. "Uniform convergence of the empirical cumulative distribution function under informative selection from a finite population." Bernoulli 18 (4) 1361 - 1385, November 2012. https://doi.org/10.3150/11-BEJ369

Information

Published: November 2012
First available in Project Euclid: 12 November 2012

zbMATH: 1329.62053
MathSciNet: MR2995800
Digital Object Identifier: 10.3150/11-BEJ369

Rights: Copyright © 2012 Bernoulli Society for Mathematical Statistics and Probability

JOURNAL ARTICLE
25 PAGES


SHARE
Vol.18 • No. 4 • November 2012
Back to Top