Adaptive function estimation in nonparametric regression with one-sided errors

Abstract

We consider the model of nonregular nonparametric regression where smoothness constraints are imposed on the regression function $f$ and the regression errors are assumed to decay with some sharpness level at their endpoints. The aim of this paper is to construct an adaptive estimator for the regression function $f$. In contrast to the standard model where local averaging is fruitful, the nonregular conditions require a substantial different treatment based on local extreme values. We study this model under the realistic setting in which both the smoothness degree $\beta>0$ and the sharpness degree $\mathfrak{a}\in(0,\infty)$ are unknown in advance. We construct adaptation procedures applying a nested version of Lepski’s method and the negative Hill estimator which show no loss in the convergence rates with respect to the general $L_{q}$-risk and a logarithmic loss with respect to the pointwise risk. Optimality of these rates is proved for $\mathfrak{a}\in(0,\infty)$. Some numerical simulations and an application to real data are provided.