The Annals of Statistics

Estimation of a function under shape restrictions. Applications to reliability

L. Reboul

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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.

Article information

Ann. Statist., Volume 33, Number 3 (2005), 1330-1356.

First available in Project Euclid: 1 July 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G07: Density estimation 62G08: Nonparametric regression 62N01: Censored data models 62N02: Estimation

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


Reboul, L. Estimation of a function under shape restrictions. Applications to reliability. Ann. Statist. 33 (2005), no. 3, 1330--1356. doi:10.1214/009053605000000138.

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