Estimation of the Derivatives of a Function in a Convolution Regression Model with Random Design

Christophe Chesneau,Maher Kachour

doi:10.1155/2015/695904

Abstract

A convolution regression model with random design is considered. We investigate the estimation of the derivatives of an unknown function, element of the convolution product. We introduce new estimators based on wavelet methods and provide theoretical guarantees on their good performances.

Highlights

We consider the convolution regression model with random design described as follows
We investigate the estimation of the derivatives of an unknown function, element of the convolution product
Let (Y1, X1), . . . , (Yn, Xn) be n i.i.d. random variables defined on a probability space (Ω, A, P), where

Summary

Introduction

We consider the convolution regression model with random design described as follows. We introduce new estimators for f(m) based on wavelet methods. Through the use of a multiresolution analysis, these methods enjoy local adaptivity against discontinuities and provide efficient estimators for a wide variety of unknown functions f(m). The second one uses the double hard thresholding technique introduced by Delyon and Juditsky [14] It does not depend on the smoothness of f(m) in its construction; it is adaptive. The obtained rates of convergence coincide with existing results for the estimation of f(m) in the 1-periodic convolution regression models (see, for instance, Chesneau [15]). Its construction follows the idea of the “NES linear wavelet estimator” introduced by Pensky and Vidakovic [16] in another regression context.

Preliminaries

Rates of Convergence

When h Is Known

Conclusion and Perspectives

Proofs