NONPARAMETRIC MAXIMUM-LIKELIHOOD-ESTIMATION BASED ON RANKED SET SAMPLES

A ranked set sample consists entirely of independently distributed ord er statistics and can occur naturally in many experimental settings, i ncluding problems in reliability. When each ranked set from which an o rder 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 ed f F(n) of a balanced ranked set sample from the cdf F is an unbiased e stimator of F and is more precise than the edf of a simple random samp le of the same size. The nonparametric maximum likelihood estimator (M LE) F of F is studied in this article. Its existence and uniqueness i s demonstrated, and a general numerical procedure is presented and is shown to converge to F. If the ranked set sample is balanced, it is s hown that the EM algorithm, with F(n) as a seed, converges to the uniq ue solution (F) of the problem's self-consistency equations; the cons istency of every iterate of the EM algorithm is also demonstrated. The modifications needed to obtain similar results in unbalanced cases ar e also discussed. Finally, the results of a simulation study are repor ted, which support the claim that the nonparametric maximum likelihood estimator, as approximated by an appropriate iterate of the EM algori thm, performs well in the unbalanced case where F(n) is inapplicable a nd performs better than F(n) in balanced cases where both estimators e xist and can be compared.