Research Article | Open Access
Christophe Chesneau, Maher Kachour, "Estimation of the Derivatives of a Function in a Convolution Regression Model with Random Design", Advances in Statistics, vol. 2015, Article ID 695904, 11 pages, 2015. https://doi.org/10.1155/2015/695904
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.
We consider the convolution regression model with random design described as follows. Let be i.i.d. random variables defined on a probability space , where, is an unknown function, is a known function, are i.i.d. random variables with common density , and are i.i.d. random variables such that and . Throughout this paper, we assume that , , and are compactly supported with , , , , , , , , is times differentiable with , is integrable and ordinary smooth (the precise definition is given by (K2) in Section 3.1), and and are independent for any . We aim to estimate the unknown function and its th derivative, denoted by , from the sample .
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 -periodic convolution regression models as those considered by, for example, Cavalier and Tsybakov , Pensky and Sapatinas , and Loubes and Marteau . In the form (1), it has been considered in Bissantz and Birke  and Birke et al.  with a deterministic design and in Hildebrandt et al.  with a random design. These last works focus on kernel methods and establish their asymptotic normality. The estimation of , more general to , is of interest to examine possible bumps and to study the convexity-concavity properties of (see, for instance, Prakasa Rao , 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 , Härdle et al. , and Vidakovic . Results on the wavelet estimation of in other regression frameworks can be found in, for example, Cai , Petsa and Sapatinas , and Chesneau .
The first part of the study is devoted to the case where , the common density of , 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 . It does not depend on the smoothness of in its construction; it is adaptive. We exhibit their rates of convergence via the mean 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 -periodic convolution regression models (see, for instance, Chesneau ).
The second part is devoted to the case where is unknown. We construct a new linear wavelet estimator using a plug-in approach for the estimation of . Its construction follows the idea of the “NES linear wavelet estimator” introduced by Pensky and Vidakovic  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 . Furthermore, let us mention that all our results are proved with only moments of order on , 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.
This section is devoted to the presentation of the considered wavelet basis, the Besov balls, and our wavelet estimation methodology.
2.1. Wavelet Basis
Let us briefly present the wavelet basis on the interval , , introduced by Cohen et al. . Let and be the initial wavelet functions of the Daubechies wavelets family db2N with (see, e.g., Daubechies ). These functions have the distinction of being compactly supported and belong to the class for . For any and , we set
With appropriated treatments at the boundaries, there exist an integer and a set of consecutive integers of cardinality proportional to (both depending on , , and ) such that, for any integer , forms an orthonormal basis of the space of squared integrable functions on ; that is, For the case and , is the smallest integer satisfying and .
For any integer and , we have the following wavelet expansion: whereAn 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.  and Mallat .
2.2. Besov Balls
We say that a function belongs to the Besov ball with , , , and if there exists a constant such that and (6) satisfy with the usual modifications if or .
2.3. Wavelet Estimation
Let be the unknown function in (1) and the considered wavelet basis taken with (to ensure that and belong to the class ). Suppose that exists with .
The first step in the wavelet estimation consists in expanding on aswhere and
The second step is the estimation of and using . The idea of the third step is to exploit 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 , providing an estimator for .
In this study, we evaluate the performance of 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 .
3. 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.
Let us recall that and are the functions in (1) and is the density of .
We formulate the following assumptions.(K1)We have for any , , and there exists a known constant such that .(K2)First of all, let us define the Fourier transform of an integrable function by The notation will be used for the complex conjugate. We have and there exist two constants, and , such that(K3)There exists a constant such that The assumptions (K1) and (K3) are standard in a nonparametric regression framework (see, for instance, Tsybakov ). Remark that we do not need for the estimation of . The assumption (K2) is the so-called “ordinary smooth case” on . It is common for the deconvolution-estimation of densities (see, e.g., Fan and Koo  and Pensky and Vidakovic ). An example of compactly supported function satisfying (K2) is . Then , , and (K2) is satisfied with and .
3.2. When Is Known
3.2.1. Linear Wavelet Estimator
We define the linear wavelet estimator bywhereand is an integer chosen a posteriori.
Proposition 1 presents an elementary property of .
Theorem 2 below investigates the performance of in terms of rates of convergence under the MISE over Besov balls.
Theorem 2. Suppose that (K1)–(K3) are satisfied and that with , , , , and . Let be defined by (14) with such that ( denotes the integer part of ).
Then there exists a constant such that
Note that the rate of convergence corresponds to the one obtained in the estimation of in the -periodic white noise convolution model with an adapted linear wavelet estimator (see, e.g., Chesneau ).
The considered estimator depends on (the smoothness parameter of ); it is not adaptive. This aspect, as well as the rate of convergence , can be improved with thresholding methods. The next paragraph is devoted to one of them: the hard thresholding method.
3.2.2. Hard Thresholding Wavelet Estimator
Suppose that (K2) is satisfied. We define the hard thresholding wavelet estimator by, where is defined by (15), is the indicator function, is a large enough constant, is the integer satisfying refers to (12),
The construction of uses the double hard thresholding technique introduced by Delyon and Juditsky  and recently improved by Chaubey et al. . The main interest of the thresholding using is to make adaptive; the construction (and performance) of 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 admits finite moments of order (with known or a known upper bound of ), relaxing the standard assumption , for any .
Further details on the constructions of hard thresholding wavelet estimators can be found in, for example, Donoho and Johnstone [26, 27], Donoho et al. [21, 28], Delyon and Juditsky , and Härdle et al. .
Theorem 3 below investigates the performance of in terms of rates of convergence under the MISE over Besov balls.
Theorem 3. Suppose that (K1)–(K3) are satisfied and that with , , , or , and . Let be defined by (19). Then there exists a constant such that
The proof of Theorem 3 is an application of a general result established by [25, Theorem 6.1]. Let us mention that corresponds to the rate of convergence obtained in the estimation of in the -periodic white noise convolution model with an adapted hard thresholding wavelet estimator (see, e.g., Chesneau ). In the case and , this rate of convergence becomes the optimal one in the minimax sense for the standard density-regression estimation problems (see Härdle et al. ).
In comparison to Theorem 2, note that(i)for the case corresponding to the homogeneous zone of Besov balls is equal to the rate of convergence attained by up to a logarithmic term,(ii)for the case corresponding to the inhomogeneous zone of Besov balls it is significantly better in terms of power.
3.3. When Is Unknown
In the case where is unknown, we propose a plug-in technique which consists in estimating in the construction of (14). This yields the linear wavelet estimator defined bywhere, is an integer chosen a posteriori, refers to (K3), and is an estimator of constructed from the random variables .
There are numerous possibilities for the choice of . For instance, can be a kernel density estimator or a wavelet density estimator (see, e.g., Donoho et al. , Härdle et al. , and Juditsky and Lambert-Lacroix ).
Theorem 4 below determines an upper bound of the MISE of .
Theorem 4. Suppose that (K1)–(K3) are satisfied, , and that with , , , , and . Let be defined by (24) with such that . Then there exists a constant such that with .
The proof follows the idea of [13, Theorem 3] and uses technical operations on Fourier transforms.
From Theorem 4,(i)if we chose and (17), we obtain Theorem 2,(ii)if and satisfy that there exist and a constant such that then, the optimal integer is such that and we obtain the following rate of convergence for : Naturally the estimation of has a negative impact on the performance of . In particular, if , then the standard density linear wavelet estimator attains the rate of convergence with and (and it is optimal in the minimax sense for ; see Härdle et al. ). With this choice, the rate of convergence for becomes . Let us mention that is not adaptive since it depends on . However, remains an acceptable first approach for the estimation of with unknown .
Conclusion and Perspectives. This study considers the estimation of from (1). According to the knowledge of 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.(i)The relaxation of the assumption (K2), perhaps by considering (K2′): there exist four constants, , , , and , such that This condition was first introduced by Delaigle and Meister  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 of through the use of a thresholding method.(iii)The extension of our results to the risk with .All these aspects need further investigations that we leave for future works.
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 .
Proof of Proposition 1. By the independence between and , , , and , we haveIt follows from (K1) and integration by parts that . Using the Fubini theorem, , (30), and the Parseval identity, we obtain Proposition 1 is proved.
Proof of Theorem 2. We expand the function on as (8) at the level . Since forms an orthonormal basis of , we getUsing Proposition 1, that are i.i.d., the inequalities for any random complex variable and , , and (K1) and (K3), we haveThe Parseval identity yieldsUsing (K2), , and a change of variables, we obtain(Let us mention that is finite thanks to .)
Combining (33), (34), and (35), we have For the integer satisfying (17), it holds thatLet us now bound the last term in (32). Since (see [9, Corollary 9.2]), we obtainOwing to (32), (37), and (38), we have Theorem 2 is proved.
Proof of Theorem 3. For , any integer , and , (a1)using arguments similar to those in Proposition 1, we obtain (a2)using (33), (34), and (35) with instead of , we have with .Thanks to (a1) and (a2), we can apply [25, Theorem 6.1] (see Appendix) with , , , , and with , , either and or and , we prove the existence of a constant such thatTheorem 3 is proved.
Proof of Theorem 4. We expand the function on as (8) at the level . Since forms an orthonormal basis of , we getUsing (see [9, Corollary 9.2]), we haveLet be (15) with and . The elementary inequality , , yieldswhere Upper Bound for . Proceeding as in (37), we getUpper Bound for . The triangular inequality gives Owing to the triangular inequality, the indicator function, (K3), , and the Markov inequality, we have Therefore where Let us now consider . For any complex random variable , we have the equality where denotes the expectation of conditionally to and and the variance of conditionally to . Thereforewhere Let us now observe that, owing to the independence of , the random variables , − conditionally to are independent. Using this property with the inequalities for any complex random variable and , , the independence between and , (K1) and (K3), we get Owing to (K2), , and a change of variables, we obtain Therefore, using