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

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.

The motivation of this problem is the deconvolution of a signal from ⋆ perturbed by noise and randomly observed. The function can represent a driving force that was applied to a physical system. Such situations naturally appear in various applied areas, as astronomy, optics, seismology, and biology. Model (1) can also be viewed as a natural extension of some 1-periodic convolution regression models as those considered by, for example, Cavalier and Tsybakov [1], Pensky and Sapatinas [2], and Loubes and Marteau [3]. In the form (1), it has been considered in Bissantz and Birke [4] and Birke et al. [5] with a deterministic design and in Hildebrandt et al. [6] with a random design. These last works focus on kernel methods and establish their asymptotic normality. The estimation of ( ) , more general to = (0) , is of interest to examine possible bumps and to study the convexity-concavity properties of (see, for instance, Prakasa Rao [7], for standard statistical models).
In this paper, we introduce new estimators for ( ) 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 ( ) . Basics on wavelet estimation can be found in, for example, Antoniadis [8], Härdle et al. [9], and Vidakovic [10]. Results on the wavelet estimation of ( ) in other regression frameworks can be found in, for example, Cai [11], Petsa and Sapatinas [12], and Chesneau [13].
The first part of the study is devoted to the case where ℎ, the common density of 1 , . . . , , is known. We develop a linear wavelet estimator and an adaptive nonlinear wavelet estimator. The second one uses the double hard thresholding technique introduced by Delyon and Juditsky [14]. It does not depend on the smoothness of ( ) in its construction; it is adaptive. We exhibit their rates of convergence via the mean 2 Advances in Statistics integrated squared error (MISE) and the assumption that ( ) belongs to Besov balls. The obtained rates of convergence coincide with existing results for the estimation of ( ) in the 1-periodic convolution regression models (see, for instance, Chesneau [15]). The second part is devoted to the case where ℎ is unknown. We construct a new linear wavelet estimator using a plug-in approach for the estimation of ℎ. Its construction follows the idea of the "NES linear wavelet estimator" introduced by Pensky and Vidakovic [16] in another regression context. Then we investigate its MISE properties when ( ) belongs to Besov balls, which naturally depend on the MISE of the considered estimator for ℎ. Furthermore, let us mention that all our results are proved with only moments of order 2 on 1 , which provides another theoretical contribution to the subject.
The remaining part of this paper is organized as follows. In Section 2 we describe some basics on wavelets and Besov balls and present our wavelet estimation methodology. Section 3 is devoted to our estimators and their performances. The proofs are carried out in Section 4.

Preliminaries
This section is devoted to the presentation of the considered wavelet basis, the Besov balls, and our wavelet estimation methodology.

Wavelet Basis.
Let us briefly present the wavelet basis on the interval [ , ], ( , ) ∈ R 2 , introduced by Cohen et al. [17]. Let and be the initial wavelet functions of the Daubechies wavelets family db2N with ≥ 1 (see, e.g., Daubechies [18]). These functions have the distinction of being compactly supported and belong to the class C for > 5 . For any ≥ 0 and ∈ Z, we set , ( ) = 2 /2 (2 − ) , With appropriated treatments at the boundaries, there exist an integer and a set of consecutive integers Λ of cardinality proportional to 2 (both depending on , , and ) such that, for any integer ℓ ≥ , forms an orthonormal basis of the space of squared integrable functions on [ , ]; that is, For the case = 0 and = 1, is the smallest integer satisfying 2 ≥ 2 and Λ = {0, . . . , 2 − 1}.
For any integer ℓ ≥ and ∈ L 2 ([ , ]), we have the following wavelet expansion: where , = ∫ ( ) , ( ) , An interesting feature of the wavelet basis is to provide sparse representation of ; only few wavelet coefficients , characterized by a high magnitude reveal the main details of . See, for example, Cohen et al. [17] and Mallat [19].

Wavelet Estimation.
Let be the unknown function in (1) and B the considered wavelet basis taken with > 5 (to ensure that and belong to the class C ). Suppose that ( ) exists with ( ) ∈ L 2 ([ , ]). The first step in the wavelet estimation consists in expanding ( ) on B as where ℓ ≥ and The second step is the estimation of ( ) , and ( ) , using ( 1 , 1 ), . . . , ( , ). The idea of the third step is to exploit Advances in Statistics 3 the sparse representation of ( ) by selecting the most interesting wavelet coefficients estimators. This selection can be of different natures (truncation, thresholding,. . .). Finally, we reconstruct these wavelet coefficients estimators on B, providing an estimator̂( ) for ( ) .
In this study, we evaluate the performance of̂( ) by studying the asymptotic properties of its MISE under the assumption that ( ) ∈ , ( ). More precisely, we aim to determine the sharpest rate of convergence such that where denotes a constant independent of .

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

Assumptions.
Let us recall that and are the functions in (1) and ℎ is the density of 1 . We formulate the following assumptions.
, and there exists a known constant First of all, let us define the Fourier transform of an integrable function by The notation ⋅ will be used for the complex conjugate.

Linear Wavelet Estimator.
We define the linear wavelet estimator̂( ) wherê and 0 is an integer chosen a posteriori. Proposition 1 presents an elementary property of̂( ) , . (15) and let ( ) , be (9). Suppose that (K1) holds. Then one has Theorem 2 below investigates the performance of̂( ) 1 in terms of rates of convergence under the MISE over Besov balls.
The considered estimator̂( ) 1 depends on (the smoothness parameter of ( ) ); it is not adaptive. This aspect, as well as the rate of convergence −2 * /(2 * +2 +2 +1) , can be improved with thresholding methods. The next paragraph is devoted to one of them: the hard thresholding method.  (15),

Hard Thresholding Wavelet Estimator. Suppose that
1 is the indicator function, > 0 is a large enough constant, 1 is the integer satisfying refers to (12), The construction of̂( ) 2 uses the double hard thresholding technique introduced by Delyon and Juditsky [14] and recently improved by Chaubey et al. [25]. The main interest of the thresholding using is to makê( ) 2 adaptive; the construction (and performance) of̂( ) 2 does not depend on the knowledge of the smoothness of ( ) . The role of the thresholding using in (20) is to relax some usual restrictions on the model. To be more specific, it enables us to only suppose that 1 admits finite moments of order 2 (with known E( 2 1 ) or a known upper bound of E( 2 1 )), relaxing the standard assumption E(| 1 | ) < ∞, for any ∈ N.
Theorem 3 below investigates the performance of̂( ) 2 in terms of rates of convergence under the MISE over Besov balls.
In comparison to Theorem 2, note that (i) for the case ≥ 2 corresponding to the homogeneous zone of Besov balls (ln / ) 2 /(2 +2 +2 +1) is equal to the rate of convergence attained bŷ( ) 1 up to a logarithmic term, (ii) for the case ∈ [1, 2) corresponding to the inhomogeneous zone of Besov balls it is significantly better in terms of power.
There are numerous possibilities for the choice ofĥ. For instance,ĥ can be a kernel density estimator or a wavelet density estimator (see, e.g., Donoho et al. [21], Härdle et al. [9], and Juditsky and Lambert-Lacroix [29]). The estimator̂( ) 3 is derived to the "NES linear wavelet estimator" introduced by Pensky and Vidakovic [16] and recently revisited in a more simple form by Chesneau [13].

Conclusion and Perspectives.
This study considers the estimation of ( ) from (1). According to the knowledge of ℎ or not, we propose wavelet methods and prove that they attain fast rates of convergence under the MISE over Besov balls. Among the perspectives of this work, we retain the following. This condition was first introduced by Delaigle and Meister [30] in a context of deconvolution-estimation of function. It implies (K2) and has the advantage to consider some functions having zeros in Fourier transform domain as numerous kinds of compactly supported functions.
(ii) The construction of an adaptive version of̂( ) 3 through the use of a thresholding method.
(iii) The extension of our results to the L risk with ≥ 1.
All these aspects need further investigations that we leave for future works.

Proofs
In this section, denotes any constant that does not depend on , , or . Its value may change from one term to another and may depend on or .

(47)
Upper Bound for 2 . Proceeding as in (37), we get Upper Bound for 1 . The triangular inequality gives 8