The Annals of Statistics

On the Estimation of a Probability Density Function by the Maximum Penalized Likelihood Method

B. W. Silverman

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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.

Article information

Ann. Statist., Volume 10, Number 3 (1982), 795-810.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62E20: Asymptotic distribution theory 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 65D10: Smoothing, curve fitting

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


Silverman, B. W. On the Estimation of a Probability Density Function by the Maximum Penalized Likelihood Method. Ann. Statist. 10 (1982), no. 3, 795--810. doi:10.1214/aos/1176345872.

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