
The fractional Korteweg-de Vries (KdV) equation generalizes the classical KdV equation by incorporating truncation effects within bounded domains, offering a flexible framework for modeling complex phenomena. This paper develops a high-order, fully discrete local discontinuous Galerkin (LDG) method with generalized alternating numerical fluxes to solve the fractional KdV equation, enhancing applicability beyond the limitations of purely alternating fluxes. An efficient finite difference scheme approximates the fractional derivatives, followed by the LDG method for solving the equation. The scheme is proven unconditionally stable and convergent. Numerical experiments confirm the method's accuracy, efficiency, and robustness, highlighting its potential for broader applications in fractional differential equations.
Citation: Yanhua Gu. High-order numerical method for the fractional Korteweg-de Vries equation using the discontinuous Galerkin method[J]. AIMS Mathematics, 2025, 10(1): 1367-1383. doi: 10.3934/math.2025063
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The fractional Korteweg-de Vries (KdV) equation generalizes the classical KdV equation by incorporating truncation effects within bounded domains, offering a flexible framework for modeling complex phenomena. This paper develops a high-order, fully discrete local discontinuous Galerkin (LDG) method with generalized alternating numerical fluxes to solve the fractional KdV equation, enhancing applicability beyond the limitations of purely alternating fluxes. An efficient finite difference scheme approximates the fractional derivatives, followed by the LDG method for solving the equation. The scheme is proven unconditionally stable and convergent. Numerical experiments confirm the method's accuracy, efficiency, and robustness, highlighting its potential for broader applications in fractional differential equations.
This paper considers the following heteroscedastic model:
Yi=f(Xi)Ui+g(Xi),i∈{1,⋯,n}. | (1.1) |
In this equation, g(x) is a known mean function, and the variance function r(x)(r(x):=f2(x)) is unknown. Both the mean function g(x) and variance function r(x) are defined on [0,1]. The random variables U1,…,Un are independent and identically distributed (i.i.d.) with E[Ui]=0 and V[Ui]=1. Furthermore, the random variable Xi is independent of Ui for any i∈{1,⋯,n}. The purpose of this paper is to estimate the mth derivative functions r(m)(x)(m∈N) from the observed data (X1,Y1),⋯,(Xn,Yn) by a wavelet method.
Heteroscedastic models are widely used in economics, engineering, biology, physical sciences and so on; see Box [1], Carroll and Ruppert [2], Härdle and Tsybakov [3], Fan and Yao [4], Quevedo and Vining [5] and Amerise [6]. For the above estimation model (1.1), the most popular method is the kernel method. Many important and interesting results of kernel estimators have been obtained by Wang et al. [7], Kulik and Wichelhaus [8] and Shen et al. [9]. However, the optimal bandwidth parameter of the kernel estimator is not easily obtained in some cases, especially when the function has some sharp spikes. Because of the good local properties in both time and frequency domains, the wavelet method has been widely used in nonparametric estimation problems; see Donoho and Johnstone [10], Cai [11], Nason et al. [12], Cai and Zhou [13], Abry and Didier [14] and Li and Zhang [15]. For the estimation problem (1.1), Kulik and Raimondo [16] studied the adaptive properties of warped wavelet nonlinear approximations over a wide range of Besov scales. Zhou et al. [17] developed wavelet estimators for detecting and estimating jumps and cusps in the mean function. Palanisamy and Ravichandran [18] proposed a data-driven estimator by applying wavelet thresholding along with the technique of sparse representation. The asymptotic normality for wavelet estimators of variance function under α−mixing condition was obtained by Ding and Chen [19].
In this paper, we focus on nonparametric estimation of the derivative function r(m)(x) of the variance function r(x). It is well known that derivative estimation plays an important and useful role in many practical applications (Woltring [20], Zhou and Wolfe, [21], Chacón and Duong [22], Wei et al.[23]). For the estimation model (1.1), a linear wavelet estimator and an adaptive nonlinear wavelet estimator for the derivative function r(m)(x) are constructed. Moreover, the convergence rates over L˜p(1≤˜p<∞) risk of two wavelet estimators are proved in Besov space Bsp,q(R) with some mild conditions. Finally, numerical experiments are carried out, where an automatic selection method is used to obtain the best parameters of two wavelet estimators. According to the simulation study, both wavelet estimators can efficiently estimate the derivative function. Furthermore, the nonlinear wavelet estimator shows better performance than the linear estimator.
This paper considers wavelet estimations of a derivative function in Besov space. Now, we first introduce some basic concepts of wavelets. Let ϕ be an orthonormal scaling function, and the corresponding wavelet function is denoted by ψ. It is well known that {ϕτ,k:=2τ/2ϕ(2τx−k),ψj,k:=2j/2ψ(2jx−k),j≥τ,k∈Z} forms an orthonormal basis of L2(R). This paper uses the Daubechies wavelet, which has a compactly support. Then, for any integer j∗, a function h(x)∈L2([0,1]) can be expanded into a wavelet series as
h(x)=∑k∈Λj∗αj∗,kϕj∗,k(x)+∞∑j=j∗∑k∈Λjβj,kψj,k(x),x∈[0,1]. | (1.2) |
In this equation, Λj={0,1,…,2j−1}, αj∗,k=⟨h,ϕj∗,k⟩[0,1] and βj,k=⟨h,ψj,k⟩[0,1].
Lemma 1.1. Let a scaling function ϕ be t-regular (i.e., ϕ∈Ct and |Dαϕ(x)|≤c(1+|x|2)−l for each l∈Z and α=0,1,…,t). If {αk}∈lp and 1≤p≤∞, there exist c2≥c1>0 such that
c12j(12−1p)‖(αk)‖p≤‖∑k∈Λjαk2j2ϕ(2jx−k)‖p≤c22j(12−1p)‖(αk)‖p. |
Besov spaces contain many classical function spaces, such as the well known Sobolev and Hölder spaces. The following lemma gives an important equivalent definition of a Besov space. More details about wavelets and Besov spaces can be found in Meyer [24] and Härdle et al. [25].
Lemma 1.2. Let ϕ be t-regular and h∈Lp([0,1]). Then, for p,q∈[1,∞) and 0<s<t, the following assertions are equivalent:
(i) h∈Bsp,q([0,1]);
(ii) {2js‖h−Pjh‖p}∈lq;
(iii) {2j(s−1p+12)‖βj,k‖p}∈lq.
The Besov norm of h can be defined by
‖h‖Bsp,q=‖(ατ,k)‖p+‖(2j(s−1p+12)‖βj,k‖p)j≥τ‖q, |
where ‖βj,k‖pp=∑k∈Λj|βj,k|p.
In this section, we will construct our wavelet estimators, and give the main theorem of this paper. The main theorem shows the convergence rates of wavelet estimators under some mild assumptions. Now, we first give the technical assumptions of the estimation model (1.1) in the following.
A1: The variance function r:[0,1]→R is bounded.
A2: For any i∈{0,…,m−1}, variance function r satisfies r(i)(0)=r(i)(1)=0.
A3: The mean function g:[0,1]→R is bounded and known.
A4: The random variable X satisfies X∼U([0,1]).
A5: The random variable U has a moment of order 2˜p(˜p≥1).
In the above assumptions, A1 and A3 are conventional conditions for nonparametric estimations. The condition A2 is used to prove the unbiasedness of the following wavelet estimators. In addition, A4 and A5 are technique assumptions, which will be used in Lemmas 4.3 and 4.5.
According to the model (1.1), our linear wavelet estimator is constructed by
ˆrlinn(x):=∑k∈Λj∗ˆαj∗,kϕj∗,k(x). | (2.1) |
In this definition, the scale parameter j∗ will be given in the following main theorem, and
ˆαj,k:=1nn∑i=1Y2i(−1)mϕ(m)j,k(Xi)−∫10g2(x)(−1)mϕ(m)j,k(x)dx. | (2.2) |
More importantly, it should be pointed out that this linear wavelet estimator is an unbiased estimator of the derivative function r(m)(x) by Lemma 4.1 and the properties of wavelets.
On the other hand, a nonlinear wavelet estimator is defined by
ˆrnonn(x):=∑k∈Λj∗ˆαj∗,kϕj∗,k(x)+j1∑j=j∗ˆβj,kI{|ˆβj,k|≥κtn}ψj,k(x). | (2.3) |
In this equation, IA denotes the indicator function over an event A, tn=2mj√lnn/n,
ˆβj,k:=1nn∑i=1(Y2i(−1)mψ(m)j,k(Xi)−wj,k)I{|Y2i(−1)mψ(m)j,k(Xi)−wj,k|≤ρn}, | (2.4) |
ρn=2mj√n/lnn, and wj,k=∫10g2(x)(−1)mψ(m)j,k(x)dx. The positive integer j∗ and j1 will also be given in our main theorem, and the constant κ will be chosen in Lemma 4.5. In addition, we adopt the following symbol: x+:=max{x,0}. A≲B denotes A≤cB for some constant c>0; A≳B means B≲A; A∼B stands for both A≲B and B≲A.
In this position, the convergence rates of two wavelet estimators are given in the following main theorem.
Main theorem For the estimation model (1.1) with the assumptions A1-A5, r(m)(x)∈Bsp,q([0,1])(p,q∈[1,∞), s>0) and 1≤˜p<∞, if {p>˜p≥1,s>0} or {1≤p≤˜p,s>1/p}.
(a) the linear wavelet estimator ˆrlinn(x) with s′=s−(1p−1˜p)+ and 2j∗∼n12s′+2m+1 satisfies
E[‖ˆrlinn(x)−r(m)(x)‖˜p˜p]≲n−˜ps′2s′+2m+1. | (2.5) |
(b) the nonlinear wavelet estimator ˆrnonn(x) with 2j∗∼n12t+2m+1 (t>s) and 2j1∼(nlnn)12m+1 satisfies
E[‖ˆrnonn(x)−r(m)(x)‖˜p˜p]≲(lnn)˜p−1(lnnn)˜pδ, | (2.6) |
where
δ=min{s2s+2m+1,s−1/p+1/˜p2(s−1/p)+2m+1}={s2s+2m+1p>˜p(2m+1)2s+2m+1s−1/p+1/˜p2(s−1/p)+2m+1p≤˜p(2m+1)2s+2m+1. |
Remark 1. Note that n−s˜p2s+1(n−(s−1/p+1/˜p)˜p2(s−1/p)+1) is the optimal convergence rate over L˜p(1≤˜p<+∞) risk for nonparametric wavelet estimations (Donoho et al. [26]). The linear wavelet estimator can obtain the optimal convergence rate when p>˜p≥1 and m=0.
Remark 2. When m=0, this derivative estimation problem reduces to the classical variance function estimation. Then, the convergence rates of the nonlinear wavelet estimator are same as the optimal convergence rates of nonparametric wavelet estimation up to a lnn factor in all cases.
Remark 3. According to main theorem (a) and the definition of the linear wavelet estimator, it is easy to see that the construction of the linear wavelet estimator depends on the smooth parameter s of the unknown derivative function r(m)(x), which means that the linear estimator is not adaptive. Compared with the linear estimator, the nonlinear wavelet estimator only depends on the observed data and the sample size. Hence, the nonlinear estimator is adaptive. More importantly, the nonlinear wavelet estimator has a better convergence rate than the linear estimator in the case of p≤˜p.
In order to illustrate the empirical performance of the proposed estimators, we produce a numerical illustration using an adaptive selection method, which is used to obtain the best parameters of the wavelet estimators. For the problem (1.1), we choose three common functions, HeaviSine, Corner and Spikes, as the mean function g(x); see Figure 1. Those functions are usually used in wavelet literature. On the other hand, we choose the function f(x) by f1(x)=3(4x−2)2e−(4x−2)2, f2(x)=sin(2πsinπx) and f3(x)=−(2x−1)2+1, respectively. In addition, we assume that the random variable U satisfies U∼N[0,1]. The aim of this paper is to estimate the derivative function r(m)(x) of the variance function r(x)(r=f2) by the observed data (X1,Y1),…,(Xn,Yn). In this section, we adopt r1(x)=[f1(x)]2, r2(x)=[f2(x)]2 and r3(x)=[f3(x)]2. For the sake of simplicity, our simulation study focuses on the derivative function r′(x)(m=1) and r(x)(m=0) by the observed data (X1,Y1),…,(Xn,Yn)(n=4096). Furthermore, we use the mean square error (MSE(ˆr(x),r(x))=1nn∑i=1(ˆr(Xi)−r(Xi))2) and the average magnitude of error (AME(ˆr(x),r(x))=1nn∑i=1|ˆr(Xi)−r(Xi)|) to evaluate the performances of the wavelet estimators separately.
For the linear and nonlinear wavelet estimators, the scale parameter j∗ and threshold value λ(λ=κtn) play important roles in the function estimation problem. In order to obtain the optimal scale parameter and threshold value of wavelet estimators, this section uses the two-fold cross validation (2FCV) approach (Nason [27], Navarro and Saumard [28]). During the first example of simulation study, we choose HeaviSine as the mean function g(x), and f1(x)=3(4x−2)2e−(4x−2)2. The estimation results of two wavelet estimators are presented by Figure 2. For the optimal scale parameter j∗ of the linear wavelet estimator, we built a collection of j∗ and j∗=1,…,log2(n)−1. The best parameter j∗ is selected by minimizing a 2FCV criterion denoted by 2FCV(j∗); see Figure 2(a). According to Figure 2(a), it is easy to see that the 2FCV(j∗) and MSE both can get the minimum value when j∗=4. For the nonlinear wavelet estimator, the best threshold value λ is also obtained by the 2FCV(λ) criterion in Figure 2(b). Meanwhile, the parameter j∗ is same as the linear estimator, and the parameter j1 is chosen as the maximum scale parameter log2(n)−1. From Figure 2(c) and 2(d), the linear and nonlinear wavelet estimators both can get a good performance with the best scale parameter and threshold value. More importantly, the nonlinear wavelet estimator shows better performance than the linear estimator.
In the following simulation study, more numerical experiments are presented to sufficiently verify the performance of the wavelet method. According to Figures 3–10, the wavelet estimators both can obtain good performances in different cases. Especially, the nonlinear wavelet estimator gets better estimation results than the linear estimator. Also, the MSE and AME of the wavelet estimators in all examples are provided by Table 1. Meanwhile, it is easy to see from Table 1 that the nonlinear wavelet estimators can have better performance than the linear estimators.
HeaviSine | Corner | Spikes | |||||||
r1 | r2 | r3 | r1 | r2 | r3 | r1 | r2 | r3 | |
MSE(ˆrlin,r) | 0.0184 | 0.0073 | 0.0071 | 0.0189 | 0.0075 | 0.0064 | 0.0189 | 0.0069 | 0.0052 |
MSE(ˆrnon,r) | 0.0048 | 0.0068 | 0.0064 | 0.0044 | 0.0070 | 0.0057 | 0.0042 | 0.0061 | 0.0046 |
MSE(ˆr′lin,r′) | 0.7755 | 0.0547 | 0.0676 | 0.7767 | 0.1155 | 0.0737 | 0.7360 | 0.2566 | 0.0655 |
MSE(ˆr′non,r′) | 0.2319 | 0.0573 | 0.0560 | 0.2204 | 0.0644 | 0.0616 | 0.2406 | 0.2868 | 0.0539 |
AME(ˆrlin,r) | 0.0935 | 0.0653 | 0.0652 | 0.0973 | 0.0667 | 0.0615 | 0.0964 | 0.0621 | 0.0550 |
AME(ˆrnon,r) | 0.0506 | 0.0641 | 0.0619 | 0.0486 | 0.0649 | 0.0583 | 0.0430 | 0.0595 | 0.0518 |
AME(ˆr′lin,r′) | 0.6911 | 0.1876 | 0.2348 | 0.7021 | 0.2686 | 0.2451 | 0.6605 | 0.4102 | 0.2320 |
AME(ˆr′non,r′) | 0.3595 | 0.1862 | 0.2125 | 0.3450 | 0.2020 | 0.2229 | 0.3696 | 0.4198 | 0.2095 |
Now, we provide some lemmas for the proof of the main Theorem.
Lemma 4.1. For the model (1.1) with A2 and A4,
E[ˆαj,k]=αj,k, | (4.1) |
E[1nn∑i=1(Y2i(−1)mψ(m)j,k(Xi)−wj,k)]=βj,k. | (4.2) |
Proof. According to the definition of ˆαj,k,
E[ˆαj,k]=E[1nn∑i=1Y2i(−1)mϕ(m)j,k(Xi)−∫10g2(x)(−1)mϕ(m)j,k(x)dx]=1nn∑i=1E[Y2i(−1)mϕ(m)j,k(Xi)]−∫10g2(x)(−1)mϕ(m)j,k(x)dx=E[Y21(−1)mϕ(m)j,k(X1)]−∫10g2(x)(−1)mϕ(m)j,k(x)dx=E[r(X1)U21(−1)mϕ(m)j,k(X1)]+2E[f(X1)U1g(X1)(−1)mϕ(m)j,k(X1)]+E[g2(X1)(−1)mϕ(m)j,k(X1)]−∫10g2(x)(−1)mϕ(m)j,k(x)dx. |
Then, it follows from A4 that
E[g2(X1)(−1)mϕ(m)j,k(X1)]=∫10g2(x)(−1)mϕ(m)j,k(x)dx. |
Using the assumption of independence between Ui and Xi,
E[r(X1)U21(−1)mϕ(m)j,k(X1)]=E[U21]E[r(X1)(−1)mϕ(m)j,k(X1)], |
E[f(X1)U1g(X1)(−1)mϕ(m)j,k(X1)]=E[U1]E[f(X1)g(X1)(−1)mϕ(m)j,k(X1)]. |
Meanwhile, the conditions V[U1]=1 and E[U1]=0 imply E[U21]=1. Hence, one gets
E[ˆαj,k]=E[r(X1)(−1)mϕ(m)j,k(X1)]=∫10r(x)(−1)mϕ(m)j,k(x)dx=(−1)m∫10r(x)ϕ(m)j,k(x)dx=∫10r(m)(x)ϕj,k(x)dx=αj,k |
by the assumption A2.
On the other hand, one takes ψ instead of ϕ, and wj,k instead of ∫10g2(x)(−1)mϕ(m)j,k(x)dx. The second equation will be proved by the similar mathematical arguments.
Lemma 4.2. (Rosenthal's inequality) Let X1,…,Xn be independent random variables such that E[Xi]=0 and E[|Xi|p]<∞. Then,
E[|n∑i=1Xi|p]≲{n∑i=1E[|Xi|p]+(n∑i=1E[|Xi|2])p2, p > 2 ,(n∑i=1E[|Xi|2])p2,1≤p≤2. |
Lemma 4.3. For the model (1.1) with A1–A5, 2j≤n and 1≤˜p<∞,
E[|ˆαj,k−αj,k|˜p]≲n−˜p22˜pmj, | (4.3) |
E[|ˆβj,k−βj,k|˜p]≲(lnnn)−˜p22˜pmj. | (4.4) |
Proof. By (4.1) and the independence of random variables Xi and Ui, one has
|ˆαj,k−αj,k|=|1nn∑i=1Y2i(−1)mϕ(m)j,k(Xi)−∫10g2(x)(−1)mϕ(m)j,k(x)dx−E[ˆαj,k]|=1n|n∑i=1(Y2i(−1)mϕ(m)j,k(Xi)−E[Y2i(−1)mϕ(m)j,k(Xi)])|=1n|n∑i=1Ai|. |
In this above equation, Ai:=Y2i(−1)mϕ(m)j,k(Xi)−E[Y2i(−1)mϕ(m)j,k(Xi)].
According to the definition of Ai, one knows that E[Ai]=0 and
E[|Ai|˜p]=E[|Y2i(−1)mϕ(m)j,k(Xi)−E[Y2i(−1)mϕ(m)j,k(Xi)]|˜p]≲E[|Y2i(−1)mϕ(m)j,k(Xi)|˜p]≲E[|(r(X1)U21+g2(X1))(−1)mϕ(m)j,k(Xi)|˜p]≲E[U2˜p1]E[|r(X1)ϕ(m)j,k(Xi)|˜p]+E[|g2(X1)ϕ(m)j,k(Xi)|˜p]. |
The assumption A5 shows E[U2˜p1]≲1. Furthermore, it follows from A1 and A3 that
E[U2˜p1]E[|r(X1)ϕ(m)j,k(X1)|˜p]≲E[|ϕ(m)j,k(X1)|˜p],E[g2˜p(X1)|ϕ(m)j,k(X1)|˜p]≲E[|ϕ(m)j,k(X1)|˜p]. |
In addition, and the properties of wavelet functions imply that
E[|ϕ(m)j,k(Xi)|˜p]=∫10|ϕ(m)j,k(x)|˜pdx=2j(˜p/2+m˜p−1)∫10|ϕ(m)(2jx−k)|˜pd(2jx−k)=2j(˜p/2+m˜p−1)||ϕ(m)||˜p˜p≲2j(˜p/2+m˜p−1). |
Hence,
E[|Ai|˜p]≲2j(˜p/2+m˜p−1). |
Especially in ˜p=2, E[|Ai|2]≲22mj.
Using Rosenthal's inequality and 2j≤n,
E[|ˆαj,k−αj,k|˜p]=1n˜pE[|n∑i=1Ai|˜p]≲{1n˜p(n∑i=1E[|Ai|˜p]+(n∑i=1E[|Ai|2])˜p2),˜p>2,1n˜p(n∑i=1E[|Ai|2])˜p2,1≤˜p≤2,≲{1n˜p(n⋅2j(˜p2+m˜p−1)+(n⋅22mj)˜p2),˜p>2,1n˜p(n⋅22mj)˜p2,1≤˜p≤2,≲n−˜p22˜pmj. |
Then, the first inequality is proved.
For the second inequality, note that
βj,k=E[1nn∑i=1(Y2i(−1)mψ(m)j,k(Xi)−wj,k)]=1nn∑i=1E[(Y2i(−1)mψ(m)j,k(Xi)−∫10g2(x)(−1)mψ(m)j,k(x)dx)]=1nn∑i=1E[Ki] |
with (4.2) and Ki:=Y2i(−1)mψ(m)j,k(Xi)−∫10g2(x)(−1)mψ(m)j,k(x)dx.
Let Bi:=KiI{|Ki|≤ρn}−E[KiI{|Ki|≤ρn}]. Then, by the definition of ˆβj,k in (2.4),
|ˆβj,k−βj,k|=|1nn∑i=1KiI{|Ki|≤ρn}−βj,k|≤1n|n∑i=1Bi|+1nn∑i=1E[|Ki|I{|Ki|>ρn}]. | (4.5) |
Similar to the arguments of Ai, it is easy to see that E[Bi]=0 and
E[|Bi|˜p]≲E[|KiI{|Ki|≤ρn}|˜p]≲E[|Ki|˜p]≲2j(˜p2+m˜p−1). |
Especially in the case of ˜p=2, one can obtain E[|Bi|2]≲22mj. On the other hand,
E[|Ki|I{|Ki|>ρn}]≲E[|Ki|⋅|Ki|ρn]=E[K21]ρn≲22mjρn=tn=2mj√lnnn. | (4.6) |
According to Rosenthal's inequality and 2j≤n,
E[|ˆβj,k−βj,k|˜p]≲1n˜pE[|n∑i=1Bi|˜p]+(tn)˜p≲{1n˜p(n∑i=1E[|Bi|˜p]+(n∑i=1E[|Bi|2])˜p2)+(tn)˜p,˜p>2,1n˜p(n∑i=1E[|Bi|2])˜p2+(tn)˜p,1≤˜p≤2,≲{1n˜p(n⋅2j(˜p2+m˜p−1)+(n⋅22mj)˜p2)+(lnnn)−˜p2⋅2˜pmj,˜p>2,1n˜p(n⋅22mj)˜p2+(lnnn)−˜p2⋅2˜pmj,1≤˜p≤2,≲(lnnn)−˜p22˜pmj. |
Then, the second inequality is proved.
Lemma 4.4. (Bernstein's inequality) Let X1,…,Xn be independent random variables such that E[Xi]=0, |Xi|<M and E[|Xi|2]:=σ2. Then, for each ν>0
P(1n|n∑i=1Xi|≥ν)≤2exp{−nν22(σ2+νM/3)}. |
Lemma 4.5. For the model (1.1) with A1–A5 and 1≤˜p<+∞, there exists a constant κ>1 such that
P(|ˆβj,k−βj,k|≥κtn)≲n−˜p. | (4.7) |
Proof. According to (4.5), one gets Ki=Y2i(−1)mψ(m)j,k(Xi)−∫10g2(x)(−1)mψ(m)j,k(x)dx, Bi=KiI{|Ki|≤ρn}−E[KiI{|Ki|≤ρn}] and
|ˆβj,k−βj,k|≤1n|n∑i=1Bi|+1nn∑i=1E[|Ki|I{|Ki|>ρn}]. |
Meanwhile, (4.6) shows that there exists c>0 such that E[|Ki|I{|Ki|>ρn}]≤ctn. Furthermore, the following conclusion is true.
{|ˆβj,k−βj,k,u|≥κtn}⊆{[1n|n∑i=1Bi|+1nn∑i=1E(|Ki|I{|Ki|>ρn})]≥κtn}⊆{1n|n∑i=1Bi|≥(κ−c)tn}. |
Note that the definition of Bi implies that |Bi|≲ρn and E[Bi]=0. Using the arguments of Lemma 4.3, E[B2i]:=σ2≲22mj. Furthermore, by Bernstein's inequality,
P(1n|n∑i=1Bi|≥(κ−c)tn)≲exp{−n(κ−c)2tn22(σ2+(κ−c)tnρn/3)}≲exp{−n(κ−c)222mj⋅lnnn2(22mj+(κ−c)⋅22mj/3)}=exp{−(lnn)(κ−c)22(1+(κ−c)/3)}=n−(κ−c)22(1+(κ−c)/3). |
Then, one can choose large enough κ such that
P(|ˆβj,k−βj,k|≥κtn)≲n−(κ−c)22(1+(κ−c)/3)≲n−˜p. |
Proof of (a): Note that
‖ˆrlinn(x)−r(m)(x)‖˜p˜p≲‖ˆrlinn(x)−Pj∗r(m)(x)‖˜p˜p+‖Pj∗r(m)(x)−r(m)(x)‖˜p˜p |
Hence,
E[‖ˆrlinn(x)−r(m)(x)‖˜p˜p]≲E[‖ˆrlinn(x)−Pj∗r(m)(x)‖˜p˜p]+‖Pj∗r(m)(x)−r(m)(x)‖˜p˜p. | (4.8) |
◼ The stochastic term E[‖ˆrlinn(x)−Pj∗r(m)(x)‖˜p˜p].
It follows from Lemma 1.1 that
E[‖ˆrlinn(x)−Pj∗r(m)(x)‖˜p˜p]=E[‖∑k∈Λj∗(ˆαj∗,k−αj∗,k)ϕj∗,k(x)‖˜p˜p]∼2j∗(12−1˜p)˜p∑k∈Λj∗E[|ˆαj∗,k−αj∗,k|˜p]. |
Then, according to (4.3), |Λj∗|∼2j∗ and 2j∗∼n12s′+2m+1, one gets
E[‖ˆrlinn(x)−Pj∗r(m)(x)‖˜p˜p]∼2j∗˜p2(2m+1)⋅n−˜p2∼n−˜ps′2s′+2m+1. | (4.9) |
◼ The bias term ‖Pj∗r(m)(x)−r(m)(x)‖˜p˜p.
When p>˜p≥1, s′=s−(1p−1˜p)+=s. Using Hölder inequality, Lemma 1.2 and r(m)∈Bsp,q([0,1]),
‖Pj∗r(m)(x)−r(m)(x)‖˜p˜p≲‖Pj∗r(m)(x)−r(m)(x)‖˜pp≲2−j∗˜ps=2−j∗˜ps′∼n−˜ps′2s′+2m+1. |
When 1≤p≤˜p and s>1p, one knows that Bsp,q([0,1])⊆Bs′˜p,∞([0,1]) and
‖Pj∗r(m)(x)−r(m)(x)‖˜p˜p≲2−j∗˜ps′∼n−˜ps′2s′+2m+1. |
Hence, the following inequality holds in both cases.
‖Pj∗r(m)(x)−r(m)(x)‖˜p˜p≲n−˜ps′2s′+2m+1. | (4.10) |
Finally, the results (4.8)–(4.10) show
E[‖ˆrlinn(x)−r(m)(x)‖˜p˜p]≲n−˜ps′2s′+2m+1. |
Proof of (b): By the definitions of ˆrlinn(x) and ˆrnonn(x), one has
‖ˆrnonn(x)−r(m)(x)‖˜p˜p≲‖ˆrlinn(x)−Pj∗r(m)(x)‖˜p˜p+‖r(m)(x)−Pj1+1r(m)(x)‖˜p˜p+‖j1∑j=j∗∑k∈Λj(ˆβj,kI{|ˆβj,k|≥κtn}−βj,k)ψj,k(x)‖˜p˜p. |
Furthermore,
E[‖ˆrnonn(x)−r(m)(x)‖˜p˜p]≲T1+T2+Q. | (4.11) |
In this above inequality,
T1:=E[‖ˆrlinn(x)−Pj∗r(m)(x)‖˜p˜p],T2:=‖r(m)(x)−Pj1+1r(m)(x)‖˜p˜p,Q:=E[‖j1∑j=j∗∑k∈Λj(ˆβj,kI{|ˆβj,k|≥κtn}−βj,k)ψj,k(x)‖˜p˜p]. |
◼ For T1. According to (4.9) and 2j∗∼n12t+2m+1 (t>s),
T1∼2j∗˜p2(2m+1)⋅n−˜p2∼n−˜pt2t+2m+1<n−˜ps2s+2m+1≤n−˜pδ. | (4.12) |
◼ For T2. Using similar mathematical arguments as (4.10), when p>˜p≥1, one can obtain T2:=‖r(m)(x)−Pj1+1r(m)(x)‖˜p˜p≲2−j1˜ps. This with 2j1∼(nlnn)12m+1 leads to
T2≲2−j1˜ps<(lnnn)˜ps2m+1≤(lnnn)˜ps2s+2m+1≤(lnnn)˜pδ. |
On the other hand, when 1≤p≤˜p and s>1p, one has Bsp,q([0,1])⊆Bs−1p+1˜p˜p,∞([0,1]) and
T2≲2−j1˜p(s−1/p+1/˜p)∼(lnnn)˜p(s−1/p+1/˜p)2m+1<(lnnn)˜p(s−1/p+1/˜p)2(s−1/p)+2m+1≤(lnnn)˜pδ. |
Therefore, for each 1≤˜p<∞,
T2≲(lnnn)˜pδ. | (4.13) |
◼ For Q. According to Hölder inequality and Lemma 1.1,
Q≲(j1−j∗+1)˜p−1j1∑j=j∗E[‖∑k∈Λj(ˆβj,kI{|ˆβj,k|≥κtn}−βj,k)ψj,k(x)‖˜p˜p]≲(j1−j∗+1)˜p−1j1∑j=j∗2j(12−1˜p)˜p∑k∈ΛjE[|ˆβj,kI{|ˆβj,k|≥κtn}−βj,k|˜p]. |
Note that
|ˆβj,kI{|ˆβj,k|≥κtn}−βj,k|˜p=|ˆβj,k−βj,k|˜pI{|ˆβj,k|≥κtn,|βj,k|<κtn2}+|ˆβj,k−βj,k|˜pI{|ˆβj,k|≥κtn,|βj,k|≥κtn2}+|βj,k|˜pI{|ˆβj,k|<κtn,|βj,k|>2κtn}+|βj,k|˜pI{|ˆβj,k|<κtn,|βj,k|≤2κtn}. |
Meanwhile,
{|ˆβj,k|≥κtn,|βj,k|<κtn2}⊆{|ˆβj,k−βj,k|>κtn2},{|ˆβj,k|<κtn,|βj,k|>2κtn}⊆{|ˆβj,k−βj,k|>κtn}⊆{|ˆβj,k−βj,k|>κtn2}. |
Then, Q can be decomposed as
Q≲(j1−j∗+1)˜p−1(Q1+Q2+Q3), | (4.14) |
where
Q1:=j1∑j=j∗2j(12−1˜p)˜p∑k∈ΛjE[|ˆβj,k−βj,k|˜pI{|ˆβj,k−βj,k|>κtn2}],Q2:=j1∑j=j∗2j(12−1˜p)˜p∑k∈ΛjE[|ˆβj,k−βj,k|˜pI{|βj,k|≥κtn2}],Q3:=j1∑j=j∗2j(12−1˜p)˜p∑k∈Λj|βj,k|˜pI{|βj,k|≤2κtn}. |
◼ For Q1. It follows from the Hölder inequality that
E[|ˆβj,k−βj,k|˜pI{|ˆβj,k−βj,k|>κtn2}]≤(E[|ˆβj,k−βj,k|2˜p])12[P(|ˆβj,k−βj,k|>κtn2)]12. |
By Lemma 4.3, one gets
E[|ˆβj,k−βj,k|2˜p]≲(lnnn)−˜p⋅22˜pmj. |
This with Lemma 4.5, |Λj|∼2j and 2j1∼(nlnn)12m+1 shows that
Q1≲j1∑j=j∗2j(12−1˜p)˜p2j⋅(lnnn)˜p22˜pmj⋅n−˜p2≲n−˜p2<n−˜pδ. | (4.15) |
◼ For Q2. One defines
2j′∼(nlnn)12s+2m+1. |
Clearly, 2j∗∼n12t+2m+1(t>s)≤2j′∼(nlnn)12s+2m+1<2j1∼(nlnn)12m+1. Furthermore, one rewrites
Q2=(j′∑j=j∗+j1∑j=j′+1)2j(12−1˜p)˜p∑k∈ΛjE[|ˆβj,k−βj,k|˜pI{|βj,k|≥κtn2}]:=Q21+Q22. | (4.16) |
◼ For Q21. By Lemma 4.3 and 2j′∼(nlnn)12s+2m+1,
Q21:=j′∑j=j∗2j(12−1˜p)˜p∑k∈ΛjE[|ˆβj,k−βj,k|˜pI{|βj,k|≥κtn2}]≤j′∑j=j∗2j(12−1˜p)˜p∑k∈ΛjE[|ˆβj,k−βj,k|˜p]≲(lnnn)˜p2j′∑j=j∗2j(2m+1)˜p2≲(lnnn)˜p22j′(2m+1)˜p2∼(lnnn)˜ps2s+2m+1≤(lnnn)˜pδ. | (4.17) |
◼ For Q22. Using Lemma 4.3, one has
Q22:=j1∑j=j′+12j(12−1˜p)˜p∑k∈ΛjE[|ˆβj,k−βj,k|˜pI{|βj,k|≥κtn2}]≲(lnnn)˜p2j1∑j=j′+12j(12−1˜p)˜p+˜pmj∑k∈ΛjI{|βj,k|≥κtn2}. |
When p>˜p≥1, by the Hölder inequality, tn=2mj√lnn/n, 2j′∼(nlnn)12s+2m+1 and Lemma 1.2, one can obtain that
Q22≲(lnnn)˜p2j1∑j=j′+12j(12−1˜p)˜p+˜pmj∑k∈Λj(|βj,k|κtn2)˜p≲j1∑j=j′+12j(12−1˜p)˜p∑k∈Λj|βj,k|˜p=j1∑j=j′+12j(12−1˜p)˜p‖βj,k‖˜p˜p≤j1∑j=j′+12j(12−1˜p)˜p⋅2j(1−˜pp)‖βj,k‖˜pp≲j1∑j=j′+12−j˜ps≲2−j′˜ps∼(lnnn)˜ps2s+2m+1≤(lnnn)˜pδ. | (4.18) |
When 1≤p≤˜p, it follows from Lemma 1.2 that
Q22≲(lnnn)˜p2j1∑j=j′+12j(12−1˜p)˜p+˜pmj∑k∈Λj(|βj,k|κtn2)p≲(lnnn)˜p−p2j1∑j=j′+12j(12−1˜p)˜p+j(˜p−p)m‖βj,k‖pp≤(lnnn)˜p−p2j1∑j=j′+12−j(sp+p2−˜p2−(˜p−p)m). | (4.19) |
Take
ϵ:=sp−˜p−p2(2m+1). |
Then, (4.19) can be rewritten as
Q22≲(lnnn)˜p−p2j1∑j=j′+12−jϵ. | (4.20) |
When ϵ>0 holds if and only if p>˜p(2m+1)2s+2m+1, δ=s2s+2m+1 and
Q22≲(lnnn)˜p−p22−j′ϵ∼(lnnn)˜ps2s+2m+1=(lnnn)˜pδ. | (4.21) |
When ϵ≤0 holds if and only if p≤˜p(2m+1)2s+2m+1, δ=s−1/p+1/˜p2(s−1/p)+2m+1. Define
2j″∼(nlnn)δs−1/p+1/˜p=(nlnn)12(s−1/p)+2m+1, |
and obviously, 2j′∼(nlnn)12s+2m+1<2j″∼(nlnn)δs−1/p+1/˜p<2j1∼(nlnn)12m+1. Furthermore, one rewrites
Q22=(j″∑j=j′+1+j1∑j=j″+1)2j(12−1˜p)˜p∑k∈ΛjE[|ˆβj,k−βj,k|˜pI{|βj,k|≥κtn2}]:=Q221+Q222. | (4.22) |
For Q221. Note that ˜p−p2+δϵs−1/p+1/˜p=˜pδ in the case of ϵ≤0. Then, by the same arguments of (4.20), one gets
Q221≲(lnnn)˜p−p2j″∑j=j′+12−jϵ≲(lnnn)˜p−p22−j″ϵ∼(lnnn)˜pδ. | (4.23) |
For Q222. The conditions 1≤p≤˜p and s>1/p imply Bsp,q([0,1])⊂Bs−1p+1˜p˜p,q([0,1]). Similar to (4.18), one obtains
Q222≲(lnnn)˜p2j1∑j=j″+12j(12−1˜p)˜p+˜pmj∑k∈Λj(|βj,k|κtn2)˜p≲j1∑j=j″+12j(12−1˜p)˜p‖βj,k‖˜p˜p≲j1∑j=j″+12j(12−1˜p)˜p⋅2−j(s−1˜p+12)˜p≲2−j″(s−1p+1˜p)˜p∼(lnnn)˜pδ. | (4.24) |
Combining (4.18), (4.21), (4.23) and (4.24),
Q22≲(lnnn)˜pδ. |
This with (4.16) and (4.17) shows that
Q2≲(lnnn)˜pδ. | (4.25) |
◼ For Q3. According to the definition of 2j′, one can write
Q3=(j′∑j=j∗+j1∑j=j′+1)2j(12−1˜p)˜p∑k∈Λj|βj,k|˜pI{|βj,k|≤2κtn}:=Q31+Q32. |
◼ For Q31. It is easy to see that
Q31:=j′∑j=j∗2j(12−1˜p)˜p∑k∈Λj|βj,k|˜pI{|βj,k|≤2κtn}≤j′∑j=j∗2j(12−1˜p)˜p∑k∈Λj(2κtn)˜p≲(lnnn)˜p2⋅2(2m+1)j′˜p2∼(lnnn)˜ps2s+2m+1≤(lnnn)˜pδ. |
◼ For Q32. One rewrites Q32=j1∑j=j′+12j(12−1˜p)˜p∑k∈Λj|βj,k|˜pI{|βj,k|≤2κtn}. When p>˜p≥1, using the Hölder inequality and Lemma 1.2,
Q32≤j1∑j=j′+12j(12−1˜p)˜p∑k∈Λj|βj,k|˜p≲2−j′˜ps∼(lnnn)˜ps2s+2m+1≤(lnnn)˜pδ. |
When 1≤p≤˜p, one has
Q32≤j1∑j=j′+12j(12−1˜p)˜p∑k∈Λj|βj,k|˜p(2κtn|βj,k|)˜p−p≲(lnnn)˜p−p2j1∑j=j′+12j(12−1˜p)˜p+j(˜p−p)m‖βj,k‖pp≤(lnnn)˜p−p2j1∑j=j′+12−j(sp+p2−˜p2−(˜p−p)m)=(lnnn)˜p−p2j1∑j=j′+12−jϵ. |
For the case of ϵ>0, one can easily obtain that δ=s2s+2m+1 and
Q32≲(lnnn)˜p−p22−j′ϵ∼(lnnn)˜ps2s+2m+1=(lnnn)˜pδ. |
When ϵ≤0, δ=s−1/p+1/˜p2(s−1/p)+2m+1. Moreover, by the definition of 2j″, one rewrites
Q32=(j″∑j=j′+1+j1∑j=j″+1)2j(12−1˜p)˜p∑k∈Λj|βj,k|˜pI{|βj,k|≤2κtn}:=Q321+Q322. |
Note that
Q321≲(lnnn)˜p−p2j″∑j=j′+12−jϵ≲(lnnn)˜p−p22−j″ϵ∼(lnnn)˜pδ. |
On the other hand, similar to the arguments of (4.24), one has
Q322≤j1∑j=j″+12j(12−1˜p)˜p∑k∈Λj|βj,k|˜p=j1∑j=j″+12j(12−1˜p)˜p‖βj,k‖˜p˜p≲(lnnn)˜pδ. |
Therefore, in all of the above cases,
Q3≲(lnnn)˜pδ. | (4.26) |
Finally, combining the above results (4.14), (4.15), (4.25) and (4.26), one gets
Q≲(j1−j∗+1)˜p−1(lnnn)˜pδ≲(lnn)˜p−1(lnnn)˜pδ. |
This with (4.11)–(4.13) shows
E[‖ˆrnonn(x)−r(m)(x)‖˜p˜p]≲(lnn)˜p−1(lnnn)˜pδ. |
This paper considers wavelet estimations of the derivatives r(m)(x) of the variance function r(x) in a heteroscedastic model. The upper bounds over L˜p(1≤˜p<∞) risk of the wavelet estimators are discussed under some mild assumptions. The results show that the linear wavelet estimator can obtain the optimal convergence rate in the case of p>˜p≥1. When p≤˜p, the nonlinear wavelet estimator has a better convergence rate than the linear estimator. Moreover, the nonlinear wavelet estimator is adaptive. Finally, some numerical experiments are presented to verify the good performances of the wavelet estimators.
We would like to thank the reviewers for their valuable comments and suggestions, which helped us to improve the quality of the manuscript. This paper is supported by the Guangxi Natural Science Foundation (No. 2022JJA110008), National Natural Science Foundation of China (No. 12001133), Center for Applied Mathematics of Guangxi (GUET), and Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation.
All authors declare that they have no conflicts of interest.
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HeaviSine | Corner | Spikes | |||||||
r1 | r2 | r3 | r1 | r2 | r3 | r1 | r2 | r3 | |
MSE(ˆrlin,r) | 0.0184 | 0.0073 | 0.0071 | 0.0189 | 0.0075 | 0.0064 | 0.0189 | 0.0069 | 0.0052 |
MSE(ˆrnon,r) | 0.0048 | 0.0068 | 0.0064 | 0.0044 | 0.0070 | 0.0057 | 0.0042 | 0.0061 | 0.0046 |
MSE(ˆr′lin,r′) | 0.7755 | 0.0547 | 0.0676 | 0.7767 | 0.1155 | 0.0737 | 0.7360 | 0.2566 | 0.0655 |
MSE(ˆr′non,r′) | 0.2319 | 0.0573 | 0.0560 | 0.2204 | 0.0644 | 0.0616 | 0.2406 | 0.2868 | 0.0539 |
AME(ˆrlin,r) | 0.0935 | 0.0653 | 0.0652 | 0.0973 | 0.0667 | 0.0615 | 0.0964 | 0.0621 | 0.0550 |
AME(ˆrnon,r) | 0.0506 | 0.0641 | 0.0619 | 0.0486 | 0.0649 | 0.0583 | 0.0430 | 0.0595 | 0.0518 |
AME(ˆr′lin,r′) | 0.6911 | 0.1876 | 0.2348 | 0.7021 | 0.2686 | 0.2451 | 0.6605 | 0.4102 | 0.2320 |
AME(ˆr′non,r′) | 0.3595 | 0.1862 | 0.2125 | 0.3450 | 0.2020 | 0.2229 | 0.3696 | 0.4198 | 0.2095 |
HeaviSine | Corner | Spikes | |||||||
r1 | r2 | r3 | r1 | r2 | r3 | r1 | r2 | r3 | |
MSE(ˆrlin,r) | 0.0184 | 0.0073 | 0.0071 | 0.0189 | 0.0075 | 0.0064 | 0.0189 | 0.0069 | 0.0052 |
MSE(ˆrnon,r) | 0.0048 | 0.0068 | 0.0064 | 0.0044 | 0.0070 | 0.0057 | 0.0042 | 0.0061 | 0.0046 |
MSE(ˆr′lin,r′) | 0.7755 | 0.0547 | 0.0676 | 0.7767 | 0.1155 | 0.0737 | 0.7360 | 0.2566 | 0.0655 |
MSE(ˆr′non,r′) | 0.2319 | 0.0573 | 0.0560 | 0.2204 | 0.0644 | 0.0616 | 0.2406 | 0.2868 | 0.0539 |
AME(ˆrlin,r) | 0.0935 | 0.0653 | 0.0652 | 0.0973 | 0.0667 | 0.0615 | 0.0964 | 0.0621 | 0.0550 |
AME(ˆrnon,r) | 0.0506 | 0.0641 | 0.0619 | 0.0486 | 0.0649 | 0.0583 | 0.0430 | 0.0595 | 0.0518 |
AME(ˆr′lin,r′) | 0.6911 | 0.1876 | 0.2348 | 0.7021 | 0.2686 | 0.2451 | 0.6605 | 0.4102 | 0.2320 |
AME(ˆr′non,r′) | 0.3595 | 0.1862 | 0.2125 | 0.3450 | 0.2020 | 0.2229 | 0.3696 | 0.4198 | 0.2095 |