Maximum likelihood estimation of ordered multinomial parameters

The pool adjacent violator algorithm Ayer et al.(1955, The Annals of Mathematical Statistics, 26, 641.647) has long been known to give the maximum likelihood estimator of a series of ordered binomial parameters, based on an independent observation from each distribution (see Barlow et al., 1972, Statistical Inference under Order Restrictions, Wiley, New York). This result has immediate application to estimation of a survival distribution based on current survival status at a set of monitoring times.This paper considers an extended problem of maximum likelihood estimation of a series of ordered multinomial parameters, where ordered means that pj1 is less than pj2 that is less than pjk for each j with j from 1 to m minus 1. The data consist of k independent observations where Xi has a multinomial distribution with probability parameter pi and known index ni greater than or equal to 1.By making use of variants of the pool adjacent violator algorithm, we obtain a simple algorithm to compute the maximum likelihood estimator of p1, ..., pk, and demonstrate its convergence. The results are applied to nonparametric maximum likelihood estimation of the sub.distribution functions associated with a survival time random variable with competing risks when only current status data are available (Jewell et al. 2003, Biometrika, 90, 183.197).