Consistent estimation of survival functions under uniform stochastic ordering; the k-sample case

Let S1,S2,…,Sk be survival functions of life distributions. They are said to be uniformly stochastically ordered, S1≤usoS2≤uso⋯≤usoSk, if Si/Si+1 is a survival function for 1≤i≤k−1. The nonparametric maximum likelihood estimators of the survival functions subject to this ordering constraint are known to be inconsistent in general. Consistent estimators were developed only for the case of k=2. In this paper we provide consistent estimators in the k-sample case, with and without censoring. In proving consistency, we needed to develop a new algorithm for isotonic regression that may be of independent interest.