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 p(i)= (p(1i),p(2i),.,p(mi)) for 1 <or=i <ro=k, where ordered means that p(j1) <or=p(j2) <or=<or=p(jk) for each j with 1 <or=j <or=m-1. The data consist of k independent observations X(1),., X(k) where X(i) has a multinomial distribution with probability parameter p(i) and known index n(i)\geq 1. By making use of variants of the pool adjacent violator algorithm, we obtain a simple algorithm to compute the maximum likelihood estimator of p(1),., p(k), 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).