## 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 χ_{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 logcondens.mode.

Original language | English (US) |
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Pages (from-to) | 2950-2976 |

Number of pages | 27 |

Journal | Annals of Statistics |

Volume | 47 |

Issue number | 5 |

DOIs | |

State | Published - 2019 |

### Bibliographical note

Funding Information:Received December 2016; revised October 2018. 1Supported in part by NSF Grants DMS-1712664, DMS-1104832 and a University of Minnesota Grant-In-Aid grant. 2Supported in part by NSF Grants DMS-1104832 and DMS-1566514, NI-AID Grant 2R01 AI291968-04 and by the Isaac Newton Institute for Mathematical Sciences program Statistical Scalability, EPSRC Grant Number LNAG/036 RG91310. MSC2010 subject classifications. Primary 62G07; secondary 62G15, 62G10, 62G20. Key words and phrases. Mode, empirical processes, likelihood ratio, pivot, convex optimization, log-concave, shape constraints.

Funding Information:

Acknowledgments. We owe thanks to Lutz Dümbgen for several helpful conversations. This work was partially supported by a grant to the second author from the Simons Foundation and was carried out in part during a visit to the Isaac Newton Institute.

## Keywords

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