We investigate the estimation of the derivatives of a regression function in the nonparametric regression model with random design. New wavelet estimators are developed. Their performances are evaluated via the mean integrated squared error. Fast rates of convergence are obtained for a wide class of unknown functions.

We consider the nonparametric regression model with random design described as follows. Let

In the literature, various estimation methods have been proposed and studied. The main ones are the kernel methods (see, e.g., [

In the first part, assuming that

The organization of this note is as follows. The next section describes some basics on wavelets and Besov balls. Our estimators and their rates of convergence are presented in Section

This section is devoted to the presentation of the considered wavelet basis and the Besov balls.

We set

With appropriated treatments at the boundaries, there exists an integer

We consider the following wavelet sequential definition of the Besov balls. We say that

The interest of Besov balls is to contain various kinds of homogeneous and inhomogeneous functions

In this section, we set the assumptions on the model, present our wavelet estimators, and determine their rates of convergence under the MISE over Besov balls.

We formulate the following assumptions.

We have

There exists a constant

There exists a constant

There exists a constant

We consider the wavelet basis

The definition of

This approach was initially introduced by Prakasa Rao [

Note that, for the standard case

Theorem

Suppose that

Then there exists a constant

The rate of convergence

In the rest of the study, the rate of convergence

The construction of

Theorem

Suppose that

The proof is based on a general result proved by [

In the case where

The estimator

Theorem

Suppose that

In addition, suppose that

Then there exists a constant

The first point of Theorem

If

Note that the assumption

Perspectives of this work are

to develop an adaptive wavelet estimator, as the hard thresholding one, for the estimation of

to relax assumptions on the model. Indeed, several techniques exist to relax

to consider dependent

These aspects need further investigations that we leave for a future work.

In this section,

First of all, we expand the function

Since

Observe that, for

using arguments similar to (

using arguments similar to (

Applying [

As in the proof of Theorem

Since

The triangular inequality gives

Combining (

A slight adaptation of [

Therefore, choosing

Let us now present in details [

We consider a general form of the hard thresholding wavelet estimator denoted by

two sequences of real numbers

Such that, for

(A1) any integer

(A2) there exist two constants,

Let

The author declares that there is no conflict of interests regarding the publication of this paper.

The author is thankful to the reviewers for their comments which have helped in improving the presentation.