On Estimating Distribution Functions Using Nomination Samples

Abstract

A nomination sample consists of independently distributed maxima from subsamples of a population with the same underlying distribution. Nomination sampling occurs when only the item with largest value is chosen from each of n independent subsamples. If the subsamples differ in size, then these observed order statistics are not identically distributed, so estimation schemes built on assumptions of independently and identically distributed (iid) data are most likely inappropriate. But we can exploit the structure of the data from the nomination sample by conditioning on the observed order of the independent maxima, and form a least squares estimator of the distribution function that minimizes risk with respect to squared error loss using an approach similar to one found in Ferguson, where the case for iid data is presented. The result is a product estimator that is consistent and compares favorably with the nonparametric maximum likelihood estimator proposed by Boyles and Samaniego, as indicated by graphs of mean squared error and Kolmogorov-Smirnov distance.

Key Words:

• Conditional risk
• Kolmogorov-Smirnov statistic
• Least squares estimation