Open Access
September, 1991 Approximation of Density Functions by Sequences of Exponential Families
Andrew R. Barron, Chyong-Hwa Sheu
Ann. Statist. 19(3): 1347-1369 (September, 1991). DOI: 10.1214/aos/1176348252


Probability density functions are estimated by the method of maximum likelihood in sequences of regular exponential families. This method is also familiar as entropy maximization subject to empirical constraints. The approximating families of log-densities that we consider are polynomials, splines and trigonometric series. Bounds on the relative entropy (Kullback-Leibler distance) 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.


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Andrew R. Barron. Chyong-Hwa Sheu. "Approximation of Density Functions by Sequences of Exponential Families." Ann. Statist. 19 (3) 1347 - 1369, September, 1991.


Published: September, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0739.62027
MathSciNet: MR1126328
Digital Object Identifier: 10.1214/aos/1176348252

Primary: 62G05
Secondary: 41A17 , 62B10 , 62F12

Keywords: $L_2$ approximation , exponential families , Kullback-Leibler number , Log-density estimation , minimum relative entropy estimation

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 3 • September, 1991
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