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June 2009 Limit distribution theory for maximum likelihood estimation of a log-concave density
Fadoua Balabdaoui, Kaspar Rufibach, Jon A. Wellner
Ann. Statist. 37(3): 1299-1331 (June 2009). DOI: 10.1214/08-AOS609


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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Fadoua Balabdaoui. Kaspar Rufibach. Jon A. Wellner. "Limit distribution theory for maximum likelihood estimation of a log-concave density." Ann. Statist. 37 (3) 1299 - 1331, June 2009.


Published: June 2009
First available in Project Euclid: 10 April 2009

zbMATH: 1160.62008
MathSciNet: MR2509075
Digital Object Identifier: 10.1214/08-AOS609

Primary: 62G20 , 62N01
Secondary: 62G05

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

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 3 • June 2009
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