INFERENCE FOR THE MODE OF A LOG-CONCAVE DENSITY

• 成果类型: Article
• 署名作者: Doss, Charles R.; Wellner, Jon A.
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities; University of Washington; University of Washington Seattle
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1770
• 发表日期: 2019
• 页码: 2950-2976
• 关键词: maximum-likelihood-estimation; nonparametric-estimation; bandwidth selection; global rates; CONVERGENCE; regression; ratio; approximation; nonexistence; functionals

摘要

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(2)(1) 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 logcondens.mode.