Maximum likelihood estimation of ordered multinomial parameters

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 P-i (p(1i), P-2i,..., p(mi)) for 1 less than or equal to i less than or equal to k, where ordered means that p(j1) less than or equal to p(j2) less than or equal to ... less than or equal to P-jk for each j with 1 less than or equal to j < 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) ≥ 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).