Nonparametric maximum likelihood estimation of survival functions with a general stochastic ordering and its dual

SUMMARY This paper discusses estimation of survival functions under an arbitrary partial stochas- tic ordering of the underlying populations. This technique is especially useful when data are not easily acquired from one or more of the populations, but data from the other populations are available. Estimation of the ordered survival functions entails use of a Fenchel duality theorem that results in an algorithm to identify and resolve violations of the underlying ordering. The algorithm produces estimates which converge to the maximum likelihood estimates of the survival functions given the underlying ordering. The method is illustrated by an example. In life testing of components and in clinical trials, estimation of a survival function P( t) is often of prime interest. If the functional form of the survival function is unknown, nonparametric methods such as the product-limit estimator (Kaplan & Meier, 1958) are needed. Johansen (1978) showed that the product-limit estimator is a maximum likelihood estimator in the class of all distributions under a generalized maximum likelihood framework (Kiefer & Wolfowitz, 1956), since the generalized maximum likelihood esti- mates for continuous survival functions place probability only on those time points at which observations occur. Suppose, however, that one desires to estimate a number of survival functions when a partial stochastic ordering exists among them and under censoring. For example, in a clinical trial of two treatments for a rare debilitating disease, let PI, P2 and P3 be the survival functions respectively for the disease-free group, and for the disease groups under the two treatments. The well-documented survival function P1 can be used to generate better estimates for P2 and P3 by using the order restrictions PI(t) ? P2(t) and PI(t) - P3(t). Brunk et al. (1966) developed the maximum likelihood estimators for two stochastically ordered distributions for uncensored independent random samples. Dykstra