Abstract: Comparing two univariate distributions based on independent samples from them is a fundamental problem in statistics, with applications in a variety of scientific disciplines. In many situations, we might hypothesize that the two distributions are stochastically ordered, meaning that samples from one distribution tend to be larger than those from the other. One type of stochastic order is the likelihood ratio order, in which the ratio of the density functions of the two distributions is monotone nondecreasing. In this article, we derive and study the nonparametric maximum likelihood estimator of the individual distribution functions and the ratio of their densities under the likelihood ratio order. Our work applies to discrete distributions, continuous distributions, and mixed continuous-discrete distributions. We demonstrate convergence in distribution of the estimator in certain cases, and illustrate our results using numerical experiments and an analysis of a biomarker for predicting bacterial infection in children with systemic inflammatory response syndrome.

Key words and phrases: Biomarker evaluation, density ratio, monotonicity constraint, odds ratio, ordinal dominance curve, shape-constrained inference.