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September, 1982 On the Estimation of a Probability Density Function by the Maximum Penalized Likelihood Method
B. W. Silverman
Ann. Statist. 10(3): 795-810 (September, 1982). DOI: 10.1214/aos/1176345872


A class of probability density estimates can be obtained by penalizing the likelihood by a functional which depends on the roughness of the logarithm of the density. The limiting case of the estimates as the amount of smoothing increases has a natural form which makes the method attractive for data analysis and which provides a rationale for a particular choice of roughness penalty. The estimates are shown to be the solution of an unconstrained convex optimization problem, and mild natural conditions are given for them to exist. Rates of consistency in various norms and conditions for asymptotic normality and approximation by a Gaussian process are given, thus breaking new ground in the theory of maximum penalized likelihood density estimation.


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B. W. Silverman. "On the Estimation of a Probability Density Function by the Maximum Penalized Likelihood Method." Ann. Statist. 10 (3) 795 - 810, September, 1982.


Published: September, 1982
First available in Project Euclid: 12 April 2007

zbMATH: 0492.62034
MathSciNet: MR663433
Digital Object Identifier: 10.1214/aos/1176345872

Primary: 62G05
Secondary: 46E35 , 62E20 , 65D10

Keywords: asymptotic normality , consitency , convex optimization existence and uniqueness , data analysis , Gaussian process , penalized likelihood , Probability density estimate , rates , ‎reproducing kernel Hilbert ‎space , roughness penalty , smoothing , Sobolev space , strong approximation

Rights: Copyright © 1982 Institute of Mathematical Statistics


Vol.10 • No. 3 • September, 1982
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