_{1}

^{*}

We study the following model:
. The aim is to estimate the distribution of
*X* when only
are observed. In the classical model, the distribution of
is assumed to be known, and this is often considered as an important drawback of this simple model. Indeed, in most practical applications, the distribution of the errors cannot be perfectly known. In this paper, the author will construct wavelet estimators and analyze their asymptotic mean integrated squared error for additive noise models under certain dependent conditions, the strong mixing case, the
*β*-mixing case and the
*ρ*-mixing case. Under mild conditions on the family of wavelets, the estimator is shown to be
-consistent and fast rates of convergence have been established.

In practical situations, direct data are not always available. One of the classical models is described as follows:

where

Sometimes, this problem can be circumvented by repeated observations of the same variable of interest, each time with an independent error. This is the model of panel data (see for example Li and Vuong [

Odiachi and Prieve [

In this paper, we extend Geng and Wang [

The organization of the paper is as follows. Assumptions on the model are presented in Section 2. Section 3 is devoted to our linear wavelet estimator and a general result. Applications are set in Section 5, while technical proofs are collected in Section 6.

The Fourier transform of

It is well known that

One can easily find an example

which is the Laplace density and

We consider an orthonormal wavelet basis generated by dilations and translations of a father Daubechies-type wavelet and a mother Daubechies-type wavelet of the family db2N (see [

With appropriated treatments at the boundaries, there exists an integer

forms an orthonormal basis of

where

with the usual modifications if

Tsybakov [

where

Such an estimator is standard in nonparametric estimation via wavelets. For a survey on wavelet linear estimators in various density models, we refer to [

In 1999, Pensky and Vidakovic [

Our work is related to the paper of Geng and Wang [

The main result of the paper is the upper bound for the mean integrated square error of the wavelet estimator

We refer to [

Theorem 3.1. Consider

a) there exists constants

b) for any

Let

Naturally, the rate of convergence in Theorem 4.1 is obtained to be as sharp as possible.

The three following subsections investigate separately the strong mixing case, the r-mixing case and the b-mixing case, which occur in a large variety of applications.

We define the m-th strong mixing coefficient of

where

Applications on strong mixing can be found in [

Proposition 4.1. Consider the strong mixing case as defined above. Suppose that there exist two constants

then

Let

where

Proposition 4.2. Consider the r-mixing case as defined above. Furthermore, there exist two constants

then

Let

where the supremum is taken over all finite partitions

Full details can be found in e.g. [

Proposition 4.3. Consider the b mixing case as defined above. Furthermore, there exist two constants

then

In this section, we investigate the results of Section 3 under the assumptions of Section 4.

Moreover, C denotes any constant that does not depend on l, k and n.

Proof of Theorem 3.1. Since we set

Following the lines of Geng and Wang [

where

and

on the other hand, it follows from the stationarity of

where

For upper bound of

By (6) and inequality obtained in Lemma 6 in [

Therefore

It follows from (5) that

Therefore, combining (7) to (11), we obtain

On the other hand, as we define

It follows from (13) and (14) and the assumption on

Now the proof of Theorem 3.1 is complete.

Proof of Proposition 5.1. We apply the Davydov inequality for strongly mixing processes (see [

Since we have

therefore

Now the proof is finished by (14), (15) and (16).

Proof of Proposition 5.2. Applying the covariance inequality for r-mixing processes (see Doukahn [

Hence by the same technique we use in (8), we obtain

Proof of Proposition 5.3. Since

where b is a function such that

N.Hosseinioun, (2016) Wavelet-Based Density Estimation in Presence of Additive Noise under Various Dependence Structures. Advances in Pure Mathematics,06,7-15. doi: 10.4236/apm.2016.61002