The Annals of Statistics

Limit distribution theory for maximum likelihood estimation of a log-concave density

Fadoua Balabdaoui, Kaspar Rufibach, and Jon A. Wellner

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Abstract

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form f0=exp ϕ0 where ϕ0 is a concave function on ℝ. The pointwise limiting distributions depend on the second and third derivatives at 0 of Hk, the “lower invelope” of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of ϕ0=log f0 at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M(f0) and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.

Article information

Source
Ann. Statist., Volume 37, Number 3 (2009), 1299-1331.

Dates
First available in Project Euclid: 10 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1239369023

Digital Object Identifier
doi:10.1214/08-AOS609

Mathematical Reviews number (MathSciNet)
MR2509075

Zentralblatt MATH identifier
1160.62008

Subjects
Primary: 62N01: Censored data models 62G20: Asymptotic properties
Secondary: 62G05: Estimation

Keywords
Asymptotic distribution integral of Brownian motion invelope process log-concave density estimation lower bounds maximum likelihood mode estimation nonparametric estimation qualitative assumptions shape constraints strongly unimodal unimodal

Citation

Balabdaoui, Fadoua; Rufibach, Kaspar; Wellner, Jon A. Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Statist. 37 (2009), no. 3, 1299--1331. doi:10.1214/08-AOS609. https://projecteuclid.org/euclid.aos/1239369023


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