Translator Disclaimer
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

Download Citation

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

Rights: Copyright © 1982 Institute of Mathematical Statistics

JOURNAL ARTICLE
14 PAGES


SHARE
Vol.10 • No. 3 • September, 1982
Back to Top