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

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

24 Aug 2007arXiv:0708.3400links table onlyarchive 2025-07-28

Fadoua Balabdaoui, Kaspar Rufibach, Jon A. Wellner

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We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form f₀=expφ₀ 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=logf₀ 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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