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September, 1982 Consistency of Two Nonparametric Maximum Penalized Likelihood Estimators of the Probability Density Function
V. K. Klonias
Ann. Statist. 10(3): 811-824 (September, 1982). DOI: 10.1214/aos/1176345873

Abstract

We study the consistency properties of a nonparametric estimator $f_n$ of a density function $f$ on the real line, which is known as the "first MPLE of Good and Gaskins," and which is obtained by maximizing the likelihood functional multiplied by the roughness penality $\exp\{- \alpha \int (f'/f)^2 f\}$ with $\alpha > 0$. Under modest assumptions on the density function $f$, and letting $\alpha = \alpha_n \rightarrow \infty$ and $\alpha_n/n \rightarrow 0$ a.s. as $n \rightarrow \infty$ we demonstrate the a.s. convergence of $f_n$ to $f$, with rates, in the Hellinger, $L_1, L_2, \sup_{\mathbb{R}}$ and Sobolev norms, as well as in integrated mean absolute deviation. Finally, the corresponding estimator for $f$ supported on the half-line, is derived and the computational feasibility as well as the consistency properties of the estimator are indicated.

Citation

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V. K. Klonias. "Consistency of Two Nonparametric Maximum Penalized Likelihood Estimators of the Probability Density Function." Ann. Statist. 10 (3) 811 - 824, September, 1982. https://doi.org/10.1214/aos/1176345873

Information

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

zbMATH: 0492.62035
MathSciNet: MR663434
Digital Object Identifier: 10.1214/aos/1176345873

Subjects:
Primary: 62G05
Secondary: 40A30 , 41A25 , 60F15 , 60F25 , 62E10 , 62G10

Keywords: exponential spline function , Fisher information functional , likelihood functional , maximum penalized likelihood method , Nonparametric density estimation , nonparametric density estimator with positive support , rates of convergence , strong consistency

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 3 • September, 1982
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