Estimation of a function under shape restrictions. Applications to reliability

Abstract

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.