Adaptive function estimation in nonparametric regression with one-sided errors

TL;DR

This paper develops an adaptive estimation method for nonparametric regression with one-sided errors, handling unknown smoothness and error sharpness, achieving optimal convergence rates under realistic nonregular conditions.

Contribution

It introduces a novel adaptive estimator using Lepski's method and the negative Hill estimator for nonregular regression with unknown smoothness and error sharpness.

Findings

Achieves optimal convergence rates for the estimator.

Demonstrates no loss in rates under unknown parameters.

Provides numerical simulations and real data application.

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.