Estimating Stochastically Ordered Survival Functions via Geometric Programming

Many procedures have been proposed to compute nonparametric maximum likelihood estimators (NPMLEs) of survival functions under stochastic ordering constraints. However, each of them is only applicable to a specific type of stochastic ordering constraint and censoring, and is often hard to implement. In this article, we describe a general and flexible method based on geometric programming for computing the NPMLEs from right- or interval-censored data. 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 real data examples to illustrate its generality in handling different types of ordering constraints and censoring. We also conduct simulation studies to examine its numerical performance for various sample sizes. Supplemental materials including technical details, computer code, and data files are available online.