Asymptotics of the discrete log-concave maximum likelihood estimator and related applications

TL;DR

This paper investigates the asymptotic properties of the log-concave maximum likelihood estimator for discrete distributions, establishing consistency, asymptotic theory, and practical confidence interval construction, with applications to pandemic data.

Contribution

It provides the first detailed asymptotic analysis of the discrete log-concave MLE, including confidence interval methods, and demonstrates practical implementation with real data.

Findings

MLE is strongly consistent for discrete log-concave pmfs

Asymptotic distributions enable confidence interval construction

Practical tools available via R package logcondiscr

Abstract

The assumption of log-concavity is a flexible and appealing nonparametric shape constraint in distribution modelling. In this work, we study the log-concave maximum likelihood estimator (MLE) of a probability mass function (pmf). We show that the MLE is strongly consistent and derive its pointwise asymptotic theory under both the well- and misspecified setting. Our asymptotic results are used to calculate confidence intervals for the true log-concave pmf. Both the MLE and the associated confidence intervals may be easily computed using the R package logcondiscr. We illustrate our theoretical results using recent data from the H1N1 pandemic in Ontario, Canada.