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February, 1995 Maximum Smoothed Likelihood Density Estimation for Inverse Problems
P. P. B. Eggermont, V. N. LaRiccia
Ann. Statist. 23(1): 199-220 (February, 1995). DOI: 10.1214/aos/1176324463


We consider the problem of estimating a pdf $f$ from samples $X_1, X_2, \ldots, X_n$ of a random variable with pdf $\mathscr{K}f$, where $\mathscr{K}$ is a compact integral operator. We employ a maximum smoothed likelihood formalism inspired by a nonlinearly smoothed version of the EMS algorithm of Silverman, Jones, Wilson and Nychka. We show that this nonlinearly smoothed algorithm is itself an EM algorithm, which helps explain the strong convergence properties of the algorithm. For the case of (standard) density estimation, that is, the case where $\mathscr{K}$ is the identity, the method yields the standard kernel density estimators. The maximum smoothed likelihood density estimation technique is a regularization technique. We prove an inequality which implies the stability and convergence of the regularization method for the large sample asymptotic problem. Under minimal assumptions it also implies the a.s. convergence of the finite sample density estimate via a uniform version of the strong law of large numbers. Under extra regularity conditions we get a.s. convergence rates via a uniform version of the law of the iterated logarithm (under stronger conditions than usual).


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P. P. B. Eggermont. V. N. LaRiccia. "Maximum Smoothed Likelihood Density Estimation for Inverse Problems." Ann. Statist. 23 (1) 199 - 220, February, 1995.


Published: February, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0822.62025
MathSciNet: MR1331664
Digital Object Identifier: 10.1214/aos/1176324463

Primary: 62G07
Secondary: 62G20 , 65R30

Keywords: Almost sure convergence , integral equation of the first kind , maximum smoothed likelihood , Nonparametric density estimation , regularization , smoothing

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 1 • February, 1995
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