Nonparametric maximum likelihood estimation of the distribution function using ranked-set sampling

Kvam and Samaniego (J Am Stat Assoc 89: 526–537, 1994) derived an estimator that they billed as the nonparametric maximum likelihood estimator (MLE) of the distribution function based on a ranked-set sample. However, we show here that the likelihood used by Kvam and Samaniego (1994) is different from the probability of seeing the observed sample under perfect rankings. By appealing to results on order statistics from a discrete distribution, we write down a likelihood that matches the probability of seeing the observed sample. We maximize this likelihood by using the EM algorithm, and we show that the resulting MLE avoids certain unintuitive behavior exhibited by the Kvam and Samaniego (1994) estimator. We find that the new MLE outperforms both the Kvam and Samaniego (1994) estimator and the unbiased estimator due to Stokes and Sager (J Am Stat Assoc 83: 374– 381, 1988) in terms of integrated mean squared error under perfect rankings.