Biostatistics (2004), 5, 2, pp. 291-306
Biostatistics Vol. 5 No. 2 © Oxford University Press 2004; all rights reserved.
Maximum likelihood estimation of ordered multinomial parameters
Division of Biostatistics, School of Public Health, University of California, Berkeley, USA
Department of Biostatistics, School of Public Health, University of Michigan, USA
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 
for 
, where ordered means that 
for each 
with 
. The data consist of 
independent observations 
where 
has a multinomial distribution with probability parameter 
and known index 
. By making use of variants of the pool adjacent violator algorithm, we obtain a simple algorithm to compute the maximum likelihood estimator of 
, 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).
Keywords: Competing risks; Current status data; Isotonic regression; Nonparametric maximum likelihood estimation