Inference for the mode of a logconcave density
Abstract
We study a likelihood ratio test for the location of the mode of a logconcave density. Our test is based on comparison of the loglikelihoods corresponding to the unconstrained maximum likelihood estimator of a logconcave 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 logconcave 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+.
 Publication:

arXiv eprints
 Pub Date:
 November 2016
 arXiv:
 arXiv:1611.10348
 Bibcode:
 2016arXiv161110348D
 Keywords:

 Mathematics  Statistics Theory;
 62G07 (primary) 62G15;
 62G10;
 62G20 (secondary)
 EPrint:
 61 pages, 4 figures