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https://hdl.handle.net/2142/20798
Description
Title
Density estimation with Kullback-Leibler loss
Author(s)
Sheu, Chyong-Hwa
Issue Date
1990
Doctoral Committee Chair(s)
Barron, Andrew
Department of Study
Statistics
Discipline
Statistics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Statistics
Language
eng
Abstract
Probability density functions are estimated by the method of maximum likelihood in sequences of regular exponential families. The approximation families of log-densities that we consider are polynomials, splines, and trigonometric series. Bounds on the relative entropy (Kullback-Leibler number) between the true density and the estimator are obtained and rates of convergence are established for log-density functions assumed to have square integrable derivatives. The relative entropy risk between true probability density function and the estimator is shown to converge to zero at a desired rate. The idea is to select n samples from the true distribution and choose the estimator which is the maximum posterior likelihood estimator in certain regular m-parameter exponential families, given that a Gaussian distribution is the prior on the parameter space. The implications for universal source coding and portfolio selection are discussed.
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