Estimating Stochastically Ordered Survival Functions via Geometric Programming

Many procedures have been proposed to compute the nonparametric maximum likelihood estimates (NPMLEs) of survival functions under various stochastic ordering constraints. Each of the existing procedures is applicable only to a specific type of stochastic order constraint and often hard to implement. In this paper, we describe a method for computing the NPMLEs of survival functions, based on geometric programming, that is applicable to more general constraints and easy to implement. To this end, we show that the monotonicity properties of the likelihood function and the stochastic ordering constraints considered in the literature allow us to reformulate the estimation problem as a geometric program (GP), a special type of mathematical optimization problem, which can be transformed to a convex optimization problem, and then solved globally and efficiently. We apply this GP-based method to the oropharynx cancer data in Kalbfleisch and Prentice (1980) to illustrate its computational merit as well as its generality in handling stochastic ordering constraints.