Nonparametric Maximum Likelihood Estimation Based on Ranked Set Samples

Abstract

A ranked set sample consists entirely of independently distributed order statistics and can occur naturally in many experimental settings, including problems in reliability. When each ranked set from which an order statistic is drawn is of the same size, and when the statistic of each fixed order is sampled the same number of times, the ranked set sample is said to be balanced. Stokes and Sager have shown that the edf F _n of a balanced ranked set sample from the cdf F is an unbiased estimator of F and is more precise than the edf of a simple random sample of the same size. The nonparametric maximum likelihood estimator (MLE) F of F is studied in this article. Its existence and uniqueness is demonstrated, and a general numerical procedure is presented and is shown to converge to F. If the ranked set sample is balanced, it is shown that the EM algorithm, with F _n as a seed, converges to the unique solution (F) of the problem's self-consistency equations; the consistency of every iterate of the EM algorithm is also demonstrated. The modifications needed to obtain similar results in unbalanced cases are also discussed. Finally, the results of a simulation study are reported, which support the claim that the nonparametric maximum likelihood estimator, as approximated by an appropriate iterate of the EM algorithm, performs well in the unbalanced case where F _n is inapplicable and performs better than F _n in balanced cases where both estimators exist and can be compared.

Key Words:

• Consistency
• EM algorithm
• Life testing
• Order statistics