Estimating a Distribution Function Based on Nomination Sampling

原文摘要

Nomination sampling is a sampling process in which every observation is the maximum of a random sample from some population. Assuming that all samples are taken from a single underlying distribution F, data may be viewed as consisting of pairs (Xi , Ki ), where Ki is the size of the ith sample and, given Ki = ki, Xi is distributed according to F ki. Willemain (1980), who discussed nomination sampling in the context of health care delivery, proposed an estimator for the median of F under the assumption that Ki = N, a fixed integer. In this article, the assumption of a fixed sample size N from each population is relaxed; with K taken as random, the problem of nonparametric estimation of the distribution function F is considered. The nonparametric maximum likelihood estimator of F is derived, its consistency is demonstrated, and its asymptotic behavior as a stochastic process is identified. Conditions are given under which these asymptotic results hold for nonrandom K. A by-product of this development is the consistency of Willemain's estimator of the median. Several applications are considered. For example, nomination sampling arises naturally in certain reliability experiments; the applicability of the derived estimator in problems involving designed redundancy is noted. A detailed analysis of data on 34 yearly maximum floods of the Nidd River is presented, and the estimator of the underlying flood distribution F is displayed.

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