Assumption-Free Nonparametric MLE of the Distribution Function Using Ranked-Set Sampling

Music City Center 

We study nonparametric maximum likelihood estimation of the population distribution function based on ranked-set sampling data. In other words, the probability of seeing the observed data is maximized both over all possible distributions and over all possible ranking schemes. We find that it can be achieved by adopting a ranking strategy driven by existing observed ranks. Obtaining maximum likelihood estimators becomes complicated when there are ties across ranking classes, but we develop a storage-intensive EM algorithm to overcome it. We find that the maximum likelihood estimator turns out not to be unique in general. However, imposing reasonable constraints leads to unique estimators with attractive properties. We compare our proposed maximum likelihood estimator to other estimators from the literature in terms of bias, variance, and consistency.

EM algorithm

Nonparametric maximum likelihood

Distribution functions

Order statistics

Ranked-set sampling

Ranking 

Section on Nonparametric Statistics