Inference for the mode of a log-concave density

Charles R Doss, Jon A. Wellner

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 log-likelihoods 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 χ1 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 logcondens.mode.

Original languageEnglish (US)
Pages (from-to)2950-2976
Number of pages27
JournalAnnals of Statistics
Volume47
Issue number5
DOIs
StatePublished - Jan 1 2019

Fingerprint

Log-concave
Confidence interval
Maximum Likelihood Estimator
Constrained Estimation
Interval Methods
Interval
Likelihood Ratio Statistic
Smoothing Parameter
Nuisance Parameter
Likelihood Ratio
Likelihood Ratio Test
Limiting Distribution
Null hypothesis
Monte Carlo method
Likelihood
Curvature
Converge
Software
Inference

Keywords

  • Convex optimization
  • Empirical processes
  • Likelihood ratio
  • Log-concave
  • Mode
  • Pivot
  • Shape constraints

Cite this

Inference for the mode of a log-concave density. / Doss, Charles R; Wellner, Jon A.

In: Annals of Statistics, Vol. 47, No. 5, 01.01.2019, p. 2950-2976.

Research output: Contribution to journalArticle

Doss, Charles R ; Wellner, Jon A. / Inference for the mode of a log-concave density. In: Annals of Statistics. 2019 ; Vol. 47, No. 5. pp. 2950-2976.
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