# On the Study of Multiwavelet Deconvolution Density Estimators

Authors: Xiaohui Zhou
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, multiwavelet deconvolution density estimators are presented by a linear multiwavelet expansion and a nonlinear multiwavelet expansion, respectively. Moreover, the unbiased estimation is shown, and asymptotic normality is discussed for the multiwavelet deconvolution density estimators. Finally, a numerical example is given for our discussion.


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## Body

## 1. Introduction and Preliminary

Assume thatΩ,F,P is a probability space. Y1,Y2,…,Yn are independent and identically distributed (i.i.d) random variables. They have the same model Y=X+ε, where X is a real random variable and ε denotes a random noise (error). Furthermore, X and ε are independent of each other. Let fX be the unknown probability density of X and fε be the density of ε. So the probability density fY of Y is equivalent to the convolution of fX and fε. If fε degenerates to the diac functional δ, Y reduces to be noise-free. So, approximating the density fX by an estimator f^n⋅:=f^n⋅;Y1,Y2,…,Yn can be recognized as a deconvolution problem. A wavelet estimator f^n means that f^n can be expanded by a wavelet basis. Some important work has been done, such as wavelet deconvolution estimators and asymptotic normality (seen in [1–5]). Moreover, a multiwavelet estimator implies that f^n can be denoted by a multiwavelet basis.Firstly, we introduce the concept of multiplicity multiresolution analysis (MMRA) and the expansion of multiwavelet estimators. Assume that a sequence of closed subspacesVjj∈Z in L2R satisfy the following properties:(1)
Vj⊆Vj+1(2)
∪j∈ZVj¯=L2R,∩j∈ZVj=0(3)
f∈Vj⟺f2⋅∈Vj+1(4)
There exists a function vectorΦ=ϕ1,ϕ2,…,ϕrT, such that ϕi⋅−k,i=1,…,r,k∈Z forms an orthogonal basis of the subspace V0, where Vj=closL2Rϕi,j,k=2j/2ϕi2jx−k:i=1,2,…,r,k∈Z and Φj,k=ϕ1,j,k,ϕ2,j,k,…,ϕr,j,kTFor everyj∈Z, the space Wj can be defined as an orthogonal complement of Vj into Vj+1, i.e., Vj+1=Vj⊕Wj. Then, ⊕jWj¯=L2R. There exists a function vector Ψ=ψ1,ψ2,…,ψrT∈L2Rr such that ψi⋅−k,i=1,…,r,k∈Z forms an orthogonal basis of the subspace W0, where Wj=closL2Rψi,j,k=2j/2ψi2j⋅−k,i=1,2,…,r,k∈Z. Moreover, Φ is called a multiscaling function with multiplicity r, and Ψ is called its corresponding multiwavelet.So, if a functionf∈L2R, it has the following expansion:(1)f=∑k∈ZCj0kTΦj0k+∑j≥j0∑k∈ZDjkTΦjk=∑i=1r∑k∈Zcij0kϕij0k+∑i=1r∑j≥j0∑k∈Zdijkψijk,where Cjk=c1jk,c2jk,…,crjkT,Djk=d1jk,d2jk,…,drjkT,cijk=f,ϕijk, and dijk=f,ψijk, i=1,2,…,r.Generally, assume thatPj and Qj are orthogonal projections from the space L2R to Vj and Wj, respectively. Then,(2)Pjf=∑k∈ZCjkTΦjk=∑i=1r∑k∈Zcijkϕijk,Qjf=∑k∈ZDjkTΦjk=∑i=1r∑k∈Zdijkψijk=Pj+1−Pjf.Thus,f=Pj0f+∑j=j0∞Qjf.And we also define the notationQj0,j1f=∑j=j0j1Qjf=∑j=j0j1∑i=1r∑k∈Zdijkψijk.Moreover, the Fourier transformfFT of f is defined by(3)fFTω:=∫Rfxe−iωxdx.And the inverse transform offFT is denoted by(4)fx=12π∫RfFTωeiωxdω.So, the Fourier transformΦFTω of Φx is defined as(5)ΦFTω:=∫RΦxe−iωxdx=ϕ1FTω,ϕ2FTω,…,ϕrFTωT.In this paper, choose a multiscaling functionΦ with multiplicity r satisfying the following condition:(C1)
Φ=ϕ1,ϕ2,…,ϕrT∈L2Rr and ϕiFTmω≲1+ω2−l/2 with l>1,m=0,1,2, i=1,2,…,r.Note:A≲B denotes two variables A,B satisfying A≤cB, for some constant c>0; A≳B is equivalent to B≲A, and A∼B means both A≲B and A≳B. Obviously, multiwavelets Sa4 (constructed by Shen et al.) [6] and CL (constructed by Chui and Lian) [7, 8] are examples for C1.According to condition (C1), the corresponding multiwaveletΨ=ψ1,ψ2,…,ψrT satisfies ψiFTω≲1+ω2−l/2.In fact,ϕix=1/2π∫RϕiFTωe−iωxdω. By using integration by parts,(6)ϕix≲1+x2−1.Then,(7)supx∈R∑kϕix−k≲supx∈R∑k1+x−k2−1≲1.According to multiplicity multiresolution analysis (MMRA),(8)∑kpi,j,k=∑kϕi,ϕj,1,k=∑k2∫Rϕixϕj2x−k¯dx≲1,whereΦx=2∑kPkΦ2x−k,Pk=pi,j,k1≤,i,j≤r, Ψx=2∑kQkΦ2x−k,Qk=qi,j,k1≤,i,j≤r, ϕix=2∑k∑j=1rpi,j,kϕj2x−k, and ψix=2∑k∑j=1rqi,j,kϕj2x−k.Moreover,Pijω=1/2∑kpi,j,ke−ikω are bounded. So, Qijω=1/2∑kqi,j,ke−ikω are bounded, where Qijω is constructed by Pijω(seen in [7–9]). Thus,(9)ψiFTω=∑j=1rQijω2ϕiFTω2≤∑j=1rQijω2ϕiFTω2≲1+ω22−l/2≲1+ω2−l/2.In addition, the density function fε of the random noise ε satisfies the following conditions [2]:(C2)
fεFTω≳1+ω2−β/2(C3)
fεFTmω1+ω2−β+m/2,m=0,1,2Under these two conditions, the random noiseε is said to be ill-posed.


## 2. Multiwavelet Deconvolution Density Estimators

In this section, we discuss the multiwavelet deconvolution density estimators. And some lemmas are deduced for the discussion of asymptotic normality in Section3.Similar to the discussion in [2, 3, 5], if l>β+1, the estimators can be defined as(10)c^ijk:=2j/2n∑p=1nKijϕi2jYp−k,Kijϕiy:=12π∫ReiωyϕiFTωfεFT−2jωdω.(11)d^ijk:=2j/2n∑p=1nKijψi2jYp−k,Kijψiy:=12π∫ReiωyψiFTωfεFT−2jωdω.According to equation (10), the linear multiwavelet estimator can be defined by(12)f^nx:=∑i=1r∑k∈Zc^ijkϕijk.By deducing simply, we have(13)Ec^ijk=E2j/2n∑p=1nKijϕi2jYp−k=2j/2n∑p=1nEKijϕi2jYp−k=2j/2n∑p=1n∫R12π∫Reiω2jY−kϕiFTωfεFT−2jωdωfYYdy=2j/2n2π∑p=1n∫Re−iωkϕiFTωfεFT−2jωfYFT−2jωdω=2j/2n2π∑p=1n∫Re−iωkϕiFTωfXFT−2jωdω=1n∑p=1n∫RfXx2j/2ϕi2jx−k¯dω=cijk.SoEf^nx=E∑i=1r∑k∈Zc^ijkϕijk=∑i=1r∑k∈Zcijkϕijk=PjfX. And we have the following conclusion.Theorem 1.
Assume thatc^ijk is defined in equation (10). Then, c^ijk is an unbiased estimation of cijk=∫RfXx2j/2ϕi2jx−k¯dω, i.e., Ec^ijk=cijk and Ef^nx=PjfX.The detailed proof of Theorem1 is similar to the proof of Lemma 2.2 in [1].According to the definition ofc^ijk in equation (10), the estimator f^nx=∑i=1r∑k∈Zc^ijkϕijk can be rewritten as(14)f^nx=2jn∑i=1r∑k∈Z∑p=1nKijϕi2jYp−kϕi2jx−k.To simplify the above expansion, the function(15)Ki∗x,y:=∑k∈ZKijϕix−kϕiy−kis introduced. It is similar to the discussion in [2, 3, 5]. Then,(16)f^nx=2jn∑i=1r∑p=1nKi∗2jYp,2jx.We introduceK˜i∗Y,x:=Ki∗Y,x−EKi∗Y,x. Then,(17)f^nx−Ef^nx=2jn∑i=1r∑p=1nK˜i∗2jYp,2jx.Next, the properties of the above functions are discussed in the following lemmas. Some conclusions are similar to the discussion in [2, 3, 5].Lemma 1.
Forl>β+1,β>1, the conditions (C1–C3) hold. Then, for every i=1,2,…,r, Kijϕi satisfies:(18)Kijϕi≲2jβ1+y2−1.Proof.
Assume thatζiω:=ϕiFTω/fεFT−2jω, for every i=1,2,…,r. Then,(19)Kijϕiy=12π∫Reiωyζiωdω.
Since(20)fεFTω≳1+ω2−β/2⟹fεFT−2jω≳2−jβ1+ω2−β/2and ϕiFTω≲1+ω2−l/2,(21)ζiω≲2jβ1+ω2−l−β/2⟶0,ω⟶∞.
We compute the derivative(22)ζi′ω=ϕiFT′ωfεFT4−2jω−1+2jϕiFTωfεFT′−2jωfεFT−2jω−2.
According to the conditions (C1–C3) andl>β+1,β>1,(23)ζi′ω≲1+ω2−l/21+2jω2β/2+2j1+2jω2β−1/2⟶0,ω⟶∞.
Similarly,(24)ζi″ω=ϕiFT″ωfεFT−2jω−1+2j+1ϕiFT′ωfεFT′−2jωfεFT−2jω−2−22jϕiFTωfεFT″−2jωfεFT−2jω−2+22j+1ϕiFTωfεFT′−2jω2fεFT−2jω−3.
Then,(25)ζi″ω≲1+ω2−l/21+2jω2β/2+2j+11+2jω2−β+1/21+2jω2β+22j1+2jω2−β+2/21+2jω2β+22j+11+2jω2−β+11+2jω23β/2,ζi″ω≲2jβ1+ω2−l/22−2j+ω2β/2+2−2j+ω2β−1/2+2−2j+ω2β−2/2.
So, the derivative functionsζi′ω and ζi″ω are bounded. By integration by parts,(26)Kijϕiy=12π∫Reiωyζiωdω=12πiy2∫Reiωyζi″ωdω.
Ify<1, it is obtained that(27)Kijϕiy≲∫Rζiωdω≲2jβ∫R1+ω2−l−β/2dω≲2jβ≲2jβ1+y2−1.
Ify≥1, the conclusion holds that(28)Kijϕiy≲y−2∫Rζi″ωdω≲1+|y|2−1∫Rζi″ωdω.
For everyj∈N, ℜj:=ω2−2j+ω2≥1 and ℜjc=ω2−2j+ω2<1.
Ifω∈ℜj,(29)ζi″ω≲2jβ1+ω2−l/22−2j+ω2β/2≲2jβ1+ω2−l−β/2.
Then,(30)∫ℜjζi″ωvdω≲2jβ∫ℜj1+ω2−l−β/2dω≲2jβ∫R1+ω2−l−β/2dω≲2jβ.
Ifω∈ℜjc,(31)ζi″ω≲2jβ1+ω2−l/22−2j+ω2β−2/2≲2jβ2−2j+ω2β−2/2.
Thus, forβ≥2,(32)∫ℜjcζi″ωdω≤∫ω<1ζi″ωdω≲2jβ∫ω<11+ω2β−2/2dω≲2jβ.
And if1<β<2,(33)∫ℜjcζi″ωdω≤∫ω<1ζi″ωdω≲2jβ∫01ωβ−2dω≲2jβ.
So, for everyβ>1,(34)∫ℜjcζi″ωdω≲2jβ.
Hence, for everyy≥1,(35)Kijϕiy≲2jβ1+y2−1.
The conclusion is similar to Lemma 2.1 in [2]. According to the above conclusion, we have the following lemma.Lemma 2.
Forl>β+1,β>1, under the conditions (C1–C3), define the function Fx:=1+x−1. Then, for every i=1,2,…,r, Kijϕi and Ki∗ satisfy the following:(1)
Ki∗x,y≲2jβFx−y(2)
∑k∈ZKijϕix−kKi′jϕi′y−k≲2jβFx−y(3)
∑i=1rEKi∗2jx,2jy=2−jPjfXxfX∈L2R, where PjfX is defined in equation (2)Proof.
According to the conclusion of Lemma1,(36)Kijϕix−k≲2jβ1+x−k2−1,for every i=1,2,…,r. Then,(37)supx∈R∑k∈ZKijϕix−k≲2jβ,where Z=k∈Z,x−k≥x−y/2∪k∈Z,y−k≥x−y/2 for fixed x,y∈R.
According to the definition ofKi∗x,y,(38)Ki∗x,y≤∑x−k≥x−y/2Kijϕix−kϕiy−k+∑y−k≥x−y/2Kijϕix−kϕiy−k.
On the other hand,ϕix=1/2π∫RϕiFTωeiωxdω.
According to the condition C1 and integration by parts, we have(39)ϕix≲1+x2−1=Fx,that is,(40)ϕiy−k≲Fy−k≲Fx−y.
So,(41)∑y−k≥x−y/2Kijϕix−kϕiy−k≲2jβFx−y.
Since(42)Kijϕix−k≲2jβ1+x−k2−1=2jβFx−k≤2jβFx−y2,∑x−k≥x−y/2Kijϕix−kϕiy−k≲∑k∈Z2jβFx−y2ϕiy−k.
On the other hand,ϕix≲Fx implies ∑k∈Zϕiy−k≲1. So the conclusion (2) holds.
For the conclusion (10), according to Lemma 1 and the above discussion in conclusion (2),(43)Ki′jϕi′y−k≲2jβ1+|y−k|2−1=2jβF|y−k|,∑k∈ZKijϕix−kKi′jϕi′y−k≲∑k∈ZKijϕix−k2jβFy−k≲22jβFx−y.
Next, to prove the conclusion (11),(44)EKi∗2jY,2jx=∫RKi∗2jY,2jxfYydy=∫R∑k∈ZKijϕi2jy−kϕi2jx−kfYydy.
Note thatϕix≲Fx≲1, then(45)∫R∑k∈ZKijϕi2jy−kϕi2jx−kfYydy≲2jβ∫RfYydy=2jβ.
So(46)EKi∗2jY,2jx=∑k∈Z∫RKijϕi2jy−kfYydy⋅ϕi2jx−k.
By the definition ofKijϕi in equation (10) and Fubini theorem,(47)∫RKijϕi2jy−kfYydy=∫R12π∫Reiω2jy−kϕiFTωfεFT−2jωdωfYydy=12π∫Re−iωk∫Re−iω−2jyfYydyϕiFTωfεFT−2jωdω=12π∫Re−iωkfYFT−2jωϕiFTωfεFT−2jωdω.
According to the convolutionfYFT=fXFT⋅fεFT,(48)∫RKijϕi2jy−kfYydy=12π∫RfXFT−2jωϕiFTωe−iωkdω=2−j2π∫RfXFTωϕiFT−2−jωei2−jkdω=∫RfXxϕi2jx−kdω.
The final equality is due to Plancheral formula:(49)∑i=1r2jEKi∗2jY,2jx=∑k∈Z∫RfXx2j/2ϕi2jx−kdω⋅2j/2ϕi2jx−k=∑i=1r∑k∈Z<fX,ϕijk>ϕijkx=PjfXx.
The conclusion (11) holds.Lemma 3 (see2, Theorem 3.1).
Assume thatHn⋅,⋅,n=1,2,… are symmetric functions, X1,X2,…,Xn are i.i.d random variables and Gnx,y:=EHnx,X1Hny,X1. If EHnX1,X2X1=0, EHn2X1,X2<∞n=1,2,…, and(50)limn⟶∞EGn2X1,X2+n−1EHn4X1,X4EHn2X1,X42=0.
Then,sn−1Un⟶dN0,1, where Un:=∑1≤i<j≤nHnXi,Xj and sn2:=EUn2=1/2nn−1EHn2X1,X2.


## 3. Asymptotic Normality

In this section, asymptotic normality is discussed for the linear multiwavelet deconvolution estimatorf^nx and the nonlinear estimator f^nnonx.Theorem 2.
Under the condition (C1–C3), withβ>1 and l>β+1, if fX∈LpR,p>2, then the linear estimator f^n satisfies(51)n2−j2β+1/2f^n−fX22−Ef^n−fX22⟶dN0,∑1≤i1<i2≤rsi1,i2,n2,with j⟶∞ and n−12j⟶0, where si1,i2,n2:=1/2nn−1EHi1,i2,n2Y1,Y2 and Hi1,i2,n2Yp1,Yp2 are defined by equations (55) and (62).Proof.
SinceEf^nx=Pjf, f^nx−Ef^nx∈Vj and(52)Ef^n−fX=−∑j′=j∞Qj′fX.
Sof^nx−Ef^nx and Ef^n−fX are orthogonal in L2R. We have(53)f^n−fX22=f^n−Ef^n+Ef^n−fX22=f^n−Ef^n22+Ef^n−fX22.
Assume that(54)Jn:=f^n−fX22−Ef^n−fX22=f^n−Ef^n22−Ef^n−Ef^n22,f^nx−Ef^nx2=22jn2∑i=1r∑p=1nK˜i∗2jYp,2jx2=22jn2∑i=1r∑p=1nK˜i∗2jYp,2jx2+22jn2∑1≤i1<i2≤r∑1≤p1<p2≤nK˜i1∗2jYp1,2jxK˜i2∗2jYp2,2jx+K˜i1∗2jYp2,2jxK˜i2∗2jYp1,2jx+22j+1n2∑1≤i≤r∑1≤p1<p2≤nK˜i∗2jYp1,2jxK˜i∗2jYp2,2jx+22j+1n2∑1≤i1<i2≤r∑1≤p≤nK˜i1∗2jYp,2jxK˜i2∗2jYp,2jx.
Define(55)Hi1,i2,n∗Yp1,Yp2:=∫RK˜i1∗2jYp1,2jxK˜i2∗2jYp2,2jx+K˜i1∗2jYp2,2jxK˜i2∗2jYp1,2jxdx.
According to the independence ofYpp=1n and EK˜i∗Y,x=0, we have(56)Jn=22jn2∑1≤i1<i2≤r∑1≤p1<p2≤nHi1,i2,n∗Yp1,Yp2+22j−1n2∑i=1r∑p=1nHi,i,n∗Yp,Yp−EHi,i,n∗Yp,Yp+22jn2∑1≤i≤r∑1≤p1<p2≤nHi,i,n∗Yp1,Yp2+22jn2∑1≤i1<i2≤r∑p=1nHi1,i2,n∗Yp,Yp−EHi1,i2,n∗Yp,Yp:=Jn1+Jn2+Jn3+Jn4.
According to Lemma2 and K˜i∗Y,x:=Ki∗Y,x−EKi∗Y,x, we have(57)K˜i1∗2jYp1,2jxK˜i2∗2jYp2,2jx≲22jβF2jYp1−xF2jYp2−x,varHi,i,n∗Yp,Yp≤EHi,i,n∗Yp,Yp2≲2j4β−2,varHi1,i2,n∗Yp,Yp≤EHi1,i2,n∗Yp,Yp2≲2j4β−2.
Moreover,(58)var∑i=1rHi,i,n∗Yp,Yp≤E∑i=1rHi,i,n∗Yp,Yp2≲2j4β−2,var∑1≤i1<i2≤rHi1,i2,n∗Yp,Yp≤E∑1≤i1<i2≤rHi1,i2,n∗Yp,Yp2≲2j4β−2.
By Markov’s inequality,∀ε>0,(59)PJn2≥n−12j2β+1/2ε=P∑i=1r∑p=1nHi,i,n∗Yp,Yp−EHi,i,n∗Yp,Yp≥2n2j2β−3/2ε≤P∑i=1r∑p=1nHi,i,n∗Yp,Yp−EHi,i,n∗Yp,Yp≥n2j2β−3/2ε≤var∑i=1r∑p=1nHi,i,n∗Yp,Ypn22j4β−3ε2,PJn4≥n−12j2β+1/2ε=P∑1≤i1<i2≤r∑p=1nHi1,i2,n∗Yp,Yp−EHi1,i2,n∗Yp,Yp≥n2j2β−3/2ε≤var∑1≤i1<i2≤r∑p=1nHi1,i2,n∗Yp,Ypn22j4β−3ε2.
According to the independence ofYpp=1n,(60)PJn2≥n−12j2β+1/2ε≤var∑p=1n∑i=1rHi,i,n∗Yp,Ypn22j4β−3ε2≲n2j4β−3n22j4β−3ε2=n−12jε−2,PJn4≥2j2β+1/2εn≤var∑1≤i1<i2≤r∑p=1nHi1,i2,n∗Yp,Ypn22j4β−3ε2≲n2j4β−2n22j4β−3ε2=2jnε2.
Ifn⟶∞, then for arbitrary given ε>0, n−12jε−2⟶0. Moreover, n2−j2β+1/2Jn2<ε and n2−j2β+1/2Jn4<ε,a.s. So Jn can be denoted by(61)n2−j2β+1/2Jn=n−12−j2β−3/2∑1≤i1<i2≤r∑1≤p1<p2≤nHi1,i2,n∗Yp1,Yp2+n−12−j2β−3/2∑1≤i≤r∑1≤p1<p2≤nHi,i,n∗Yp1,Yp2+o1=n−12−j2β−3/2∑1≤i1≤i2≤r∑1≤p1<p2≤nHi1,i2,n∗Yp1,Yp2+o1.
Let(62)Hi1,i2,nYp1,Yp2:=n−12−j2β−3/2+1Hi1,i2,n∗Yp1,Yp2,so(63)n2−j2β+1/2Jn=∑1≤i1≤i2≤r∑1≤p1<p2≤nHi1,i2,nYp1,Yp2+o1.
It is easy to check thatHi1,i2,nYp1,Yp2 are symmetric functions. It is similar to the work of Theorem A in [2] that EHi1,i2,n4Yp1,Yp2 and EGi1,i2,n2Yp1,Yp2 satisfy the condition of Lemma 3. According to Lemma 2 and Lemma 3,(64)si1,i2,n−1∑1≤p1<p2≤nHi1,i2,nYp1,Yp2⟶dN0,1,where si1,i2,n2:=1/2nn−1EHi1,i2,n2Y1,Y2.
Thus,(65)∑1≤p1<p2≤nHi1,i2,nYp1,Yp2⟶dN0,si1,i2,n2.
The detail discussion is similar to the proof of Theorem A in [2]. So,(66)∑1≤i1⩽i2≤r∑1≤p1<p2≤nHi1,i2,nYp1,Yp2⟶dN0,∑1≤i1⩽i2≤rsi1,i2,n2.
Consider the nonlinear multiwavelet estimator(67)f^nnonx:=∑i=1r∑k∈Zc^ij0kϕij0k+∑j=j0j1∑i=1r∑k∈Zd^ijkψijk=f^n+Qj0,j1f^nnon.Theorem 3.
Under the conditions (C1–C3), withβ>1 and l>β+1, if fX∈LpR,p>2, then the nonlinear estimator f^nnon satisfies(68)n2−j02β+1/2f^nnon−fX22−Ef^nnon−fX22⟶dN0,∑1≤i1⩽i2≤rsi1,i2,n2,with λj∼j/n2jβ,j0,i1⟶∞, and n−12j14β+4⟶0, where si1,i2,n2:=1/2nn−1EHi1,i2,n2Y1,Y2 and Hi1,i2,n2Yp1,Yp2 are defined by equations (55) and (62).
The proof is similar to Theorem B in [2].


## 4. Numerical Example

In this section, an example is given for discussing the results of multiwavelet deconvolution density estimators.Choose the modelY=X+ε. Construct the data X by the function “randn” and error data ε by the function “rand” in Matlab. That is, X∼N0,1 is a standard normal random variable and ε∼U0,1 is a uniform random variable. So fY is the convolution of fX and fε, where fXx=1/2πe−x2/2 and fεz=10<z<10else.By the formula of the convolution, we have density functionfY of random variable Y as follows:(69)fYy=∫y−1y12πe−x2/2dx,−∞<y<∞.In Figure1, random data Y is shown at the left side and its sampling number is 2048. At the right side of Figure 1, the blue dotted curve denotes the empirical density of data Y and the density of data Y is shown by the red solid curve.Figure 1
(a) DataY; (b) empirical density and density of Y.
(a)
(b)According to Theorem1 and Ef^nx=PjfX, we choose the multiwavelet Sa4 to estimate the expectation of the linear multiwavelet deconvolution density estimators. The sampling data are decomposed into 4 levels by multiwavelet transform.The densityfX of X is shown in the second row and second column of Figure 2. In the first row and first column of Figure 2, the linear multiwavelet deconvolution density estimator f^n of X is given by the black solid line and the expectation Ef^n of linear multiwavelet deconvolution density estimator f^ defined by equation (12) is shown by the red solid line. In the first row and second column of Figure 2, nonlinear multiwavelet deconvolution density estimator f^1nnon is given by the black solid line and the expectation Ef^1nnon of nonlinear multiwavelet density estimator f^1nnon is shown by the red solid line, where f^1nnon can be denoted by(70)f^1nnonx:=∑i=1r∑k∈Zc^ij0kϕij0k+∑i=1r∑k∈Zd^ij0kψij0k.Figure 2
Multiwavelet deconvolution density estimators.
(a)
(b)
(c)
(d)In the second row and first column of Figure2, nonlinear multiwavelet deconvolution density estimator f^2nnon is given by the black solid line and the expectation Ef^2nnon of nonlinear multiwavelet density estimator f^2nnon is shown by the red solid line, where f^2nnon can be denoted by(71)f^2nnonx:=∑i=1r∑k∈Zc^ij0kϕij0k+∑j=j0j0+1∑i=1r∑k∈Zd^ijkψijk.Moreover, asymptotic normality is identified by the Jarque–Bera test. The results of the J-B test are given forf^n−fX, f^1nnon−fX, and f^2nnon−fX in Table 1.Table 1
Asymptotic normality of multiwavelet density estimators.
Estimators
P value
Results of normality
f
^
n
−
f
X
0.5000
0
f
^
1
n
non
−
f
X
0.4642
0
f
^
2
n
non
−
f
X
0.5000
0
Note: this table shows the results of the J-B test for multiwavelet estimatorsf^n−fX, f^1nnon−fX, and f^2nnon−fX. If the result of Jarque–Bera test is zero, it indicates that it obeys normal distribution at significant level 0.05. If P value of the Jarque–Bera test is closer to zero, it indicates that the original assumption of normal distribution can be rejected.In Table1, all results of normality are zero, and the original assumption of normal distribution can be accepted by the P value of the Jarque–Bera test. These imply the conclusions of Theorem 2 and Theorem 3.



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*Source: 4259373-2020-04-21.xml*