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June 2000 Maximum likelihood estimation of smooth monotone and unimodal densities
P. P. B. Eggermont, V. N. LaRiccia
Ann. Statist. 28(3): 922-947 (June 2000). DOI: 10.1214/aos/1015952005

Abstract

We study the nonparametric estimation of univariate monotone and unimodal densities usingthe maximum smoothed likelihood approach. The monotone estimator is the derivative of the least concave majorant of the distribution correspondingto a kernel estimator.We prove that the mapping on distributions $\Phi$ with density $\varphi$,

$$\varphi \mapsto \text{the derivative of the least concave majorant of $\Phi},$$

is a contraction in all $L^P$ norms $(1 \leq p \leq \infty)$, and some other “distances” such as the Hellinger and Kullback–Leibler distances. The contractivity implies error bounds for monotone density estimation. Almost the same error bounds hold for unimodal estimation.

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P. P. B. Eggermont. V. N. LaRiccia. "Maximum likelihood estimation of smooth monotone and unimodal densities." Ann. Statist. 28 (3) 922 - 947, June 2000. https://doi.org/10.1214/aos/1015952005

Information

Published: June 2000
First available in Project Euclid: 12 March 2002

zbMATH: 1105.62332
MathSciNet: MR1792794
Digital Object Identifier: 10.1214/aos/1015952005

Subjects:
Primary: 62G07

Keywords: $L^1$ error bounds , contractions , least concave majorants , maximum likelihood estimation , monotone and unimodal densities

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.28 • No. 3 • June 2000
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