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June 2005 Estimation of a function under shape restrictions. Applications to reliability
L. Reboul
Ann. Statist. 33(3): 1330-1356 (June 2005). DOI: 10.1214/009053605000000138


This paper deals with a nonparametric shape respecting estimation method for U-shaped or unimodal functions. A general upper bound for the nonasymptotic $\mathbb{L}_{1}$-risk of the estimator is given. The method is applied to the shape respecting estimation of several classical functions, among them typical intensity functions encountered in the reliability field. In each case, we derive from our upper bound the spatially adaptive property of our estimator with respect to the $\mathbb{L}_{1}$-metric: it approximately behaves as the best variable binwidth histogram of the function under estimation.


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L. Reboul. "Estimation of a function under shape restrictions. Applications to reliability." Ann. Statist. 33 (3) 1330 - 1356, June 2005.


Published: June 2005
First available in Project Euclid: 1 July 2005

zbMATH: 1072.62023
MathSciNet: MR2195637
Digital Object Identifier: 10.1214/009053605000000138

Primary: 62G05
Secondary: 62G07 , 62G08 , 62N01 , 62N02

Keywords: adaptive estimation , data-driven estimator , hazard rate , nonhomogeneous Poisson process , unimodal function , U-shaped function , Variable binwidth histogram

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 3 • June 2005
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