# Nonparametric maximum likelihood estimation under a likelihood ratio order

## TL;DR

This paper develops a nonparametric maximum likelihood estimator for two distributions under the likelihood ratio order, applicable to various distribution types, with theoretical convergence results and practical applications.

## Contribution

It introduces a novel nonparametric estimator for distributions and their density ratio under likelihood ratio order, extending to discrete, continuous, and mixed cases.

## Key findings

- Estimator converges in distribution in certain cases
- Numerical experiments validate the estimator's performance
- Application to biomarker data demonstrates practical utility

## Abstract

Comparison of two univariate distributions based on independent samples from them is a fundamental problem in statistics, with applications in a wide variety of scientific disciplines. In many situations, we might hypothesize that the two distributions are stochastically ordered, meaning intuitively that samples from one distribution tend to be larger than those from the other. One type of stochastic order that arises in economics, biomedicine, and elsewhere is the likelihood ratio order, also known as the density ratio order, in which the ratio of the density functions of the two distributions is monotone non-decreasing. In this article, we derive and study the nonparametric maximum likelihood estimator of the individual distributions 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 we illustrate our results using numerical experiments and an analysis of a biomarker for predicting bacterial infection in children with systemic inflammatory response syndrome.