Nonparametric Maximum Likelihood Estimation Under a Likelihood Ratio Order

Abstract

Comparison of 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 non-decreasing. 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 we illustrate our results using numerical experiments and an analysis of a biomarker for predicting bacterial infection in children with systemic inflammatory response syndrome.

Mixture representations for likelihood ratio ordered distributions

“…Following Westling et al [37], we apply our method to data from a study by Downes et al [10]. The study enrolled children who presented symptoms of systemic inflammatory response syndrome (SIRS) at the pediatric intensive care unit of the Children's Hospital of Philadelphia between August 10, 2012 and June 9, 2016.…”

“…The data set analyzed by Westling et al [37] includes pairs {Z i , D i } of the C-reactive protein measurement Z i , in milligrams per deciliter, and the infection status D i of each patient for which this information was recorded. Although some patients had multiple SIRS episodes, these are treated as independent, because all such episodes occurred at least 30 days apart.…”

“…The most familiar stochastic orders formalize, in different ways, the intuitive idea that one random variable, or distribution, is larger than another. Monotone likelihood ratios have important implications in statistical theory [20] and feature in applications of statistics to economics and finance [30,2] as well as biology, medicine, and public health [12,5,39,37]. The likelihood ratio order is also pertinent to many problems in applied probability, stochastic processes, and operations research [32].…”

“…As motivation for this problem, and following Yu et al [39] and Westling et al [37], consider the relationship between a continuous explanatory variable Z and a binary response variable D. For example, Z could be the value of a biomarker and D could indicate the presence of a disease. In this situation, one commonly fits a generalized linear model, in which case pr(D = 1 | Z = z) is a monotonic function of z.…”

“…With a simple modification to the prior distribution, we can also calculate the posterior probability of the null hypothesis H 0 : F = G versus the alternative H 1 : F ≤ LR G, addressing a testing problem of interest in economics [30,2,3] and statistics [12,5]. Following Westling et al [37], we apply our method to the evaluation of C-reactive protein as a means of diagnosing bacterial infection in children with systemic inflammatory response syndrome.…”