The Annals of Statistics

Inference for the mode of a log-concave density

Charles R. Doss and Jon A. Wellner

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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+.

Article information

Ann. Statist., Volume 47, Number 5 (2019), 2950-2976.

Received: December 2016
Revised: October 2018
First available in Project Euclid: 3 August 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62G15: Tolerance and confidence regions 62G10: Hypothesis testing 62G20: Asymptotic properties

Mode empirical processes likelihood ratio pivot convex optimization log-concave shape constraints


Doss, Charles R.; Wellner, Jon A. Inference for the mode of a log-concave density. Ann. Statist. 47 (2019), no. 5, 2950--2976. doi:10.1214/18-AOS1770.

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Supplemental materials

  • Supplement to “Inference for the mode of a log-concave density”. In the supplement, we provide additional proofs and technical details that were omitted from the main paper.