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.

1. Introduction

We consider the convolution regression model with random design described as follows. Let (Y₁, X₁), …, (Y_n, X_n) be n i.i.d. random variables defined on a probability space

, where

()

is an unknown function,

is a known function, X₁, …, X_n are n i.i.d. random variables with common density

, and ξ₁, …, ξ_n are n i.i.d. random variables such that

and

. Throughout this paper, we assume that f, g, and h are compactly supported with supp(f) = [a, b], supp(g) = [a^′, b^′], supp(h) = [a_*, b_*],

, a < b, a^′ < b^′, a_* = a + a^′, b_* = b + b^′, f is m times differentiable with

, g is integrable and ordinary smooth (the precise definition is given by (K2) in Section 3.1), and X_v and ξ_v are independent for any v = 1, …, n. We aim to estimate the unknown function f and its mth derivative, denoted by f^(m), from the sample (Y₁, X₁), …, (Y_n, X_n).

The motivation of this problem is the deconvolution of a signal f from f⋆g perturbed by noise and randomly observed. The function g 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 f^(m), more general to f = f⁽⁰⁾, is of interest to examine possible bumps and to study the convexity-concavity properties of f (see, for instance, Prakasa Rao [7], for standard statistical models).

In this paper, 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). 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 f^(m) 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 X₁, …, X_n, 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 f^(m) in its construction; it is adaptive. We exhibit their rates of convergence via the mean integrated squared error (MISE) and the assumption that f^(m) belongs to Besov balls. 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]).

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 f^(m) 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 ξ₁, 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.

2. Preliminaries

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 [a, b],

, introduced by Cohen et al. [17]. Let ϕ and ψ be the initial wavelet functions of the Daubechies wavelets family db2N with N ≥ 1 (see, e.g., Daubechies [18]). These functions have the distinction of being compactly supported and belong to the class

for N > 5a. For any j ≥ 0 and

, we set

()

With appropriated treatments at the boundaries, there exist an integer τ and a set of consecutive integers Λ_j of cardinality proportional to 2^j (both depending on a, b, and N) such that, for any integer

()

forms an orthonormal basis of the space of squared integrable functions on [a, b]; that is,

()

For the case a = 0 and b = 1, τ is the smallest integer satisfying 2^τ ≥ 2N and Λ_j = {0, …, 2^j − 1}.

For any integer

and

, we have the following wavelet expansion:

()

where

()

An interesting feature of the wavelet basis is to provide sparse representation of u; only few wavelet coefficients d_j,k characterized by a high magnitude reveal the main details of u. See, for example, Cohen et al. [17] and Mallat [19].

2.2. Besov Balls

We say that a function

belongs to the Besov ball

with s > 0, p ≥ 1, r ≥ 1, and M > 0 if there exists a constant C > 0 such that c_j,k and d_j,k (6) satisfy

()

with the usual modifications if p = ∞ or r = ∞.

The interest of Besov balls is to contain various kinds of homogeneous and inhomogeneous functions u. See, for example, Meyer [20], Donoho et al. [21], and Härdle et al. [9].

2.3. Wavelet Estimation

Let f be the unknown function in (1) and the considered wavelet basis taken with N > 5m (to ensure that ϕ and ψ belong to the class ). Suppose that f^(m) exists with .

The first step in the wavelet estimation consists in expanding f^(m) on

()

where

and

()

The second step is the estimation of and using (Y₁, X₁), …, (Y_n, X_n). The idea of the third step is to exploit the sparse representation of f^(m) 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 f^(m).

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 ω_n such that

()

where C denotes a constant independent of n.

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.

3.1. Assumptions

Let us recall that f and g are the functions in (1) and h is the density of X₁.

We formulate the following assumptions.

(K1)
We have f^(q)(a) = f^(q)(b) = 0 for any q ∈ {0, …, m}, , and there exists a known constant C₁ > 0 such that sup⁡_x∈[a,b] | f(x)| ≤ C₁.
(K2)
First of all, let us define the Fourier transform of an integrable function u by
()
The notation will be used for the complex conjugate.
We have and there exist two constants, c₁ > 0 and δ ≥ 0, such that
()
(K3)
There exists a constant c₂ > 0 such that
()

The assumptions (K1) and (K3) are standard in a nonparametric regression framework (see, for instance, Tsybakov [22]). Remark that we do not need f(a) = f(b) = 0 for the estimation of f = f⁽⁰⁾. The assumption (K2) is the so-called “ordinary smooth case” on g. It is common for the deconvolution-estimation of densities (see, e.g., Fan and Koo [23] and Pensky and Vidakovic [24]). An example of compactly supported function g satisfying (K2) is

. Then supp(g) = [−2,2],

, and (K2) is satisfied with δ = 2 and

3.2. When h Is Known

3.2.1. Linear Wavelet Estimator

We define the linear wavelet estimator

()

where

()

and j₀ is an integer chosen a posteriori.

Proposition 1 presents an elementary property of .

Proposition 1. Let be (15) and let be (9). Suppose that (K1) holds. Then one has

()

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 M > 0, p ≥ 1, r ≥ 1, s ∈ (max⁡(1/p − 1/2,0), N), and N > 5(m + δ + 1). Let be defined by (14) with j₀ such that

()

s_* = s + min⁡(1/2 − 1/p, 0) ([a] denotes the integer part of a).

Then there exists a constant C > 0 such that

()

Note that the rate of convergence corresponds to the one obtained in the estimation of f^(m) in the 1-periodic white noise convolution model with an adapted linear wavelet estimator (see, e.g., Chesneau [15]).

The considered estimator depends on s (the smoothness parameter of f^(m)); 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

()

x ∈ [a, b], where

is defined by (15),

()

1 is the indicator function, κ > 0 is a large enough constant, j₁ is the integer satisfying

()

δ refers to (12),

()

The construction of 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 λ_j is to make adaptive; the construction (and performance) of does not depend on the knowledge of the smoothness of f^(m). The role of the thresholding using ς_j in (20) is to relax some usual restrictions on the model. To be more specific, it enables us to only suppose that ξ₁ admits finite moments of order 2 (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 [14], and Härdle et al. [9].

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 M > 0, r ≥ 1, {p ≥ 2, s ∈ (0, N)} or {p ∈ [1,2), s ∈ ((2m + 2δ + 1)/p, N)}, and N > 5(m + δ + 1). Let be defined by (19). Then there exists a constant C > 0 such that

()

The proof of Theorem 3 is an application of a general result established by [25, Theorem 6.1]. Let us mention that (ln⁡n/n)^{2s/(2s+2m+2δ+1)} corresponds to the rate of convergence obtained in the estimation of f^(m) in the 1-periodic white noise convolution model with an adapted hard thresholding wavelet estimator (see, e.g., Chesneau [15]). In the case m = 0 and δ = 0, this rate of convergence becomes the optimal one in the minimax sense for the standard density-regression estimation problems (see Härdle et al. [9]).

In comparison to Theorem 2, note that

(i)
for the case p ≥ 2 corresponding to the homogeneous zone of Besov balls (ln⁡n/n) ^{2s/(2s+2m+2δ+1)} is equal to the rate of convergence attained by up to a logarithmic term,
(ii)
for the case p ∈ [1,2) corresponding to the inhomogeneous zone of Besov balls it is significantly better in terms of power.

3.3. When h Is Unknown

In the case where h is unknown, we propose a plug-in technique which consists in estimating h in the construction of

(14). This yields the linear wavelet estimator

defined by

()

where

()

a_n = [n/2], j₂ is an integer chosen a posteriori, c₂ refers to (K3), and

is an estimator of h 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. [21], Härdle et al. [9], and Juditsky and Lambert-Lacroix [29]).

The estimator 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].

Theorem 4 below determines an upper bound of the MISE of .

Theorem 4. Suppose that (K1)–(K3) are satisfied, , and that with M > 0, p ≥ 1, r ≥ 1, s ∈ (max⁡(1/p − 1/2,0), N), and N > 5(m + δ + 1). Let be defined by (24) with j₂ such that . Then there exists a constant C > 0 such that

()

with s_* = s + min⁡(1/2 − 1/p, 0).

The proof follows the idea of [13, Theorem 3] and uses technical operations on Fourier transforms.

From Theorem 4,

(i)
if we chose and j₂ = j₀ (17), we obtain Theorem 2,
(ii)
if and h satisfy that there exist υ ∈ [0,1] and a constant C > 0 such that
()
then, the optimal integer j₂ is such that and we obtain the following rate of convergence for :
()

Naturally the estimation of h has a negative impact on the performance of

. In particular, if

, then the standard density linear wavelet estimator

attains the rate of convergence n^−υ with υ = 2s_∘/(2s_∘ + 1) and s_∘ = s^′ + min⁡(1/2 − 1/p^′, 0) (and it is optimal in the minimax sense for p^′ ≥ 2; see Härdle et al. [9]). With this choice, the rate of convergence for

becomes

. Let us mention that

is not adaptive since it depends on s. However,

remains an acceptable first approach for the estimation of f^(m) with unknown h.

Conclusion and Perspectives. This study considers the estimation of f^(m) 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.

(i)
The relaxation of the assumption (K2), perhaps by considering (K2′): there exist four constants, C₁ > 0, , η > 0, and δ ≥ 0, such that
()
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 g 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 p ≥ 1.

All these aspects need further investigations that we leave for future works.

4. Proofs

In this section, C denotes any constant that does not depend on j, k, or n. Its value may change from one term to another and may depend on ϕ or ψ.

Proof of Proposition 1. By the independence between X₁ and ξ₁, , sup⁡(f⋆g) = supp(h) = [a_*, b_*], and , we have

()

It follows from (K1) and m 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 f^(m) on as (8) at the level . Since forms an orthonormal basis of , we get

()

Using Proposition 1, (Y₁, X₁), …, (Y_n, X_n) that are i.i.d., the inequalities

for any random complex variable D and (x + y) ² ≤ 2(x² + y²),

, and (K1) and (K3), we have

()

The Parseval identity yields

()

Using (K2),

, and a change of variables, we obtain

()

(Let us mention that

is finite thanks to N > 5(m + δ + 1).)

Combining (33), (34), and (35), we have

()

For the integer j₀ satisfying (17), it holds that

()

Let us now bound the last term in (32). Since

(see [9, Corollary 9.2]), we obtain

()

Owing to (32), (37), and (38), we have

()

Theorem 2 is proved.

Proof of Theorem 3. For γ ∈ {ϕ, ψ}, any integer j ≥ τ, and k ∈ Λ_j,

(a1)
using arguments similar to those in Proposition 1, we obtain
()
(a2)
using (33), (34), and (35) with γ instead of ϕ, we have
()
with .

Thanks to (a1) and (a2), we can apply [25, Theorem 6.1] (see Appendix) with μ_n = υ_n = n, σ = m + δ, θ_γ = C_*, W_v = (Y_v, X_v),

()

and

with M > 0, r ≥ 1, either {p ≥ 2 and s ∈ (0, N)} or {p ∈ [1,2) and s ∈ (1/p, N)}, we prove the existence of a constant C > 0 such that

()

Theorem 3 is proved.

Proof of Theorem 4. We expand the function f^(m) on as (8) at the level . Since forms an orthonormal basis of , we get

()

Using

(see [9, Corollary 9.2]), we have

()

Let

be (15) with n = a_n and j = j₂. The elementary inequality (x + y) ² ≤ 2(x² + y²),

, yields

()

where

()

Upper Bound for S₂. Proceeding as in (37), we get

()

Upper Bound for S₁. 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 D, we have the equality

()

where

denotes the expectation of D conditionally to U_n and

and the variance of D conditionally to U_n. Therefore

()

where

()

Let us now observe that, owing to the independence of (Y₁, X₁), …, (Y_n, X_n), the random variables

−

conditionally to U_n are independent. Using this property with the inequalities

for any complex random variable D and (x + y) ² ≤ 2(x² + y²),

, the independence between X₁ and ξ₁, (K1) and (K3), we get

()

Owing to (K2),

, and a change of variables, we obtain

()

Therefore, using and , we obtain

()

Now, by the Hölder inequality for conditional expectations, arguments similar to (33), (34), and (35), we get

()

Hence

()

It follows from (54), (58), and (60) that

()

Putting (46), (48), and (61) together, we get

()

Combining (44), (45), and (62), we obtain the desired result; that is,

()

Theorem 4 is proved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are thankful to the reviewers for their comments which have helped in improving the presented work.

Appendix

Let us now present in detail the general result of [25, Theorem 6.1] used in the proof of Theorem 3.

We consider the wavelet basis presented in Section 2 and a general form of the hard thresholding wavelet estimator denoted by

for estimating an unknown function

from n independent random variables W₁, …, W_n:

()

where

()

, and j₁ is the integer satisfying

()

Here, we suppose that there exist

(i)
n functions q₁, …, q_n with for any i ∈ {1, …, n},
(ii)
two sequences of real numbers and satisfying lim_n→∞υ_n = ∞ and lim_n→∞μ_n = ∞

such that, for γ ∈ {ϕ, ψ},

(A1) any integer j ≥ τ and any k ∈ Λ_j,
()
(A2) there exist two constants, θ_γ > 0 and σ ≥ 0, such that, for any integer j ≥ τ and any k ∈ Λ_j,
()

Let

be (A.1) under (A1) and (A2). Suppose that

with r ≥ 1, {p ≥ 2 and s ∈ (0, N)} or {p ∈ [1,2) and s ∈ ((2σ + 1)/p, N)}. Then there exists a constant C > 0 such that

()

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

Abstract

1. Introduction

2. Preliminaries

2.1. Wavelet Basis

2.2. Besov Balls

2.3. Wavelet Estimation

3. Rates of Convergence

3.1. Assumptions

3.2. When h Is Known

3.2.1. Linear Wavelet Estimator

3.2.2. Hard Thresholding Wavelet Estimator

3.3. When h Is Unknown

4. Proofs

Conflict of Interests

Acknowledgment

Appendix

References

References

Information

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

Abstract

1. Introduction

2. Preliminaries

2.1. Wavelet Basis

2.2. Besov Balls

2.3. Wavelet Estimation

3. Rates of Convergence

3.1. Assumptions

3.2. When h Is Known

3.2.1. Linear Wavelet Estimator

3.2.2. Hard Thresholding Wavelet Estimator

3.3. When h Is Unknown

4. Proofs

Conflict of Interests

Acknowledgment

Appendix

References

References

Related

Information