Maximum Likelihood Estimation of Ordered Multinomial Parameters

Nicholas P. Jewell, Division of Biostatistics, School of Public Health, University of California, Berkeley
John D. Kalbfleisch, Dept. of Statistics & Actuarial Science, University of Waterloo, Ontario, Canada

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. By making use of variants of the pool adjacent violator algorithm, we obtain a simple algorithm to compute the maximum likelihood estimator 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., 2001).