Maximum likelihood estimation of ordered multinomial parameters

Abstract

The pool adjacent violator algorithm (Ayer et al., 1955) 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). 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 Pi = (p1i, p 2i,..., pmi) for 1 ≤ i ≤ k, where ordered means that Pj1 ≤ pj2≤ ...≤ pjk for each j with 1 ≤ j ≤m -1. The data consist of k independent observations X1,..., Xk where Xk has a multinomial distribution with probability parameter pi and known index n i ≥ 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). © Oxford University Press 2004; all rights reserved.