Inference for the mode of a log-concave density

TL;DR

This paper develops a likelihood ratio test for the mode of a log-concave density, establishing its asymptotic distribution and creating smoothing-parameter-free confidence intervals, with practical implementation and real data applications.

Contribution

It introduces a new likelihood ratio test for the mode of a log-concave density and constructs confidence intervals that are asymptotically pivotal and free of smoothing parameters.

Findings

Likelihood ratio statistic converges to a nuisance-parameter-free distribution.

New confidence intervals outperform existing methods.

Software available in R package logcondens.mode.

Abstract

We study a likelihood ratio test for the location of the mode of a log-concave density. Our test is based on comparison of the log-likelihoods corresponding to the unconstrained maximum likelihood estimator of a log-concave density and the constrained maximum likelihood estimator where the constraint is that the mode of the density is fixed, say at $m$. The constrained estimation problem is studied in detail in Doss and Wellner [2018]. Here the results of that paper are used to show that, under the null hypothesis (and strict curvature of $-\log f$ at the mode), the likelihood ratio statistic is asymptotically pivotal: that is, it converges in distribution to a limiting distribution which is free of nuisance parameters, thus playing the role of the $\chi_1^2$ distribution in classical parametric statistical problems. By inverting this family of tests we obtain new (likelihood ratio based) confidence intervals for the mode of a log-concave density $f$. These new intervals do not depend on any smoothing parameters. We study the new confidence intervals via Monte Carlo methods and illustrate them with two real data sets. The new intervals seem to have several advantages over existing procedures. Software implementing the test and confidence intervals is available in the R package \verb+logcondens.mode+.