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


Balabdaoui, F; Rufibach, K; Wellner, J A (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density. Annals of Statistics, 37(3):1299-1331.

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.

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.

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23 citations in Web of Science®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:04 Faculty of Medicine > Epidemiology, Biostatistics and Prevention Institute (EBPI)
Dewey Decimal Classification:610 Medicine & health
Language:English
Date:2009
Deposited On:18 Aug 2009 07:30
Last Modified:05 Apr 2016 13:19
Publisher:Institute of Mathematical Statistics
ISSN:0090-5364
Publisher DOI:https://doi.org/10.1214/08-AOS609
Official URL:https://www.zora.uzh.ch/20179/1/Limit_Distribution.pdf

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