### 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) |
---|---|

Pages (from-to) | 2950-2976 |

Number of pages | 27 |

Journal | Annals of Statistics |

Volume | 47 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 2019 |

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### Keywords

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

### Cite this

*Annals of Statistics*,

*47*(5), 2950-2976. https://doi.org/10.1214/18-AOS1770

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

Research output: Contribution to journal › Article

*Annals of Statistics*, vol. 47, no. 5, pp. 2950-2976. https://doi.org/10.1214/18-AOS1770

}

TY - JOUR

T1 - Inference for the mode of a log-concave density

AU - Doss, Charles R

AU - Wellner, Jon A.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Convex optimization

KW - Empirical processes

KW - Likelihood ratio

KW - Log-concave

KW - Mode

KW - Pivot

KW - Shape constraints

UR - http://www.scopus.com/inward/record.url?scp=85071029080&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071029080&partnerID=8YFLogxK

U2 - 10.1214/18-AOS1770

DO - 10.1214/18-AOS1770

M3 - Article

AN - SCOPUS:85071029080

VL - 47

SP - 2950

EP - 2976

JO - Annals of Statistics

JF - Annals of Statistics

SN - 0090-5364

IS - 5

ER -