On the Study of Multiwavelet Deconvolution Density Estimators

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.


Introduction and Preliminary
Assume that (Ω, F, P) is a probability space. Y 1 , Y 2 , . . . , Y n are independent and identically distributed (i.i.d) random variables. ey 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 f X be the unknown probability density of X and f ε be the density of ε. So the probability density f Y of Y is equivalent to the convolution of f X and f ε . If f ε degenerates to the diac functional δ, Y reduces to be noise-free. So, approximating the density f X by an estimator f n (·): � f n (·; Y 1 , Y 2 , . . . , Y n ) 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][2][3][4][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 subspaces V j j∈Z in L 2 (R) satisfy the following properties: (4) ere exists a function vector Φ � [ϕ 1 , ϕ 2 , . . . , ϕ r ] T , such that ϕ i (· − k), i � 1, . . . , r, k ∈ Z forms an orthogonal basis of the subspace V 0 , where V j � clos For every j ∈ Z, the space W j can be defined as an Moreover, Φ is called a multiscaling function with multiplicity r, and Ψ is called its corresponding multiwavelet.
So, if a function f ∈ L 2 (R), it has the following expansion: Generally, assume that P j and Q j are orthogonal projections from the space L 2 (R) to V j and W j , respectively. en, And the inverse transform of f FT is denoted by In this paper, choose a multiscaling function Φ with multiplicity r satisfying the following condition: 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 multiwavelet Ψ � [ψ 1 , ψ 2 , . . . , ψ r ] T satisfies |ψ FT i (ω)| ≲ (1 + |ω| 2 ) − (l/2) . In fact, ϕ i (x) � (1/2π) R ϕ i FT (ω)e − iωx dω. By using integration by parts, en, According to multiplicity multiresolution analysis (MMRA), [7][8][9]). us, In addition, the density function f ε of the random noise ε satisfies the following conditions [2]: Under these two conditions, the random noise ε is said to be ill-posed.

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 Section 3. Similar to the discussion in [2,3,5], if l > β + 1, the estimators can be defined as (10) According to equation (10), the linear multiwavelet estimator can be defined by By deducing simply, we have 2 Mathematical Problems in Engineering And we have the following conclusion.
e detailed proof of eorem 1 is similar to the proof of Lemma 2.2 in [1].
According to the definition of c ijk in equation (10), the estimator f n (x) � r i�1 k∈Z c ijk ϕ ijk can be rewritten as To simplify the above expansion, the function is introduced. It is similar to the discussion in [2,3,5]. en, Next, the properties of the above functions are discussed in the following lemmas. Some conclusions are similar to the discussion in [2,3,5].

Mathematical Problems in Engineering
If |y| < 1, it is obtained that If |y| ≥ 1, the conclusion holds that en, If ω ∈ R c j , us, for β ≥ 2, And if 1 < β < 2, So, for every β > 1, Hence, for every |y| ≥ 1, e conclusion is similar to Lemma 2.1 in [2]. According to the above conclusion, we have the following lemma. □ Lemma 2. For l > β + 1, β > 1, under the conditions (C1-C3), define the function F(|x|): � (1 + |x| − 1 ). en, for every i � 1, 2, . . . , r, K ij ϕ i and K * i satisfy the following: where P j f X is defined in equation (2) Proof. According to the conclusion of Lemma 1, for every i � 1, 2, . . . , r. en, According to the definition of K * i (x, y), On the other hand, According to the condition C1 and integration by parts, we have that is, So, Since On the other hand, |ϕ i (x)| ≲ F(|x|) implies k∈Z |ϕ i (y − k)| ≲ 1. So the conclusion (2) holds. For the conclusion (10), according to Lemma 1 and the above discussion in conclusion (2),

Asymptotic Normality
In this section, asymptotic normality is discussed for the linear multiwavelet deconvolution estimator f n (x) and the nonlinear estimator f non n (x).

Theorem 2.
Under the condition (C1-C3), with β > 1 and l > β + 1, if f X ∈ L p (R), (p > 2), then the linear estimator f n satisfies Mathematical Problems in Engineering with j ⟶ ∞ and n − 1 2 j ⟶ 0, where s 2 i 1 ,i 2 ,n : � (1/2)n(n − 1)EH 2 are defined by equations (55) and (62). (52) So f n (x) − Ef n (x) and Ef n − f X are orthogonal in L 2 (R). We have In Figure 1, 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.
According to eorem 1 and Ef n (x) � P j f X , we choose the multiwavelet Sa4 to estimate the expectation of the linear multiwavelet deconvolution density estimators. e sampling data are decomposed into 4 levels by multiwavelet transform.
e density f X 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) Moreover, asymptotic normality is identified by the Jarque-Bera test. e results of the J-B test are given for f n − f X , f non 1n − f X , and f non 2n − f X in Table 1.
In Table 1, 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.
ese imply the conclusions of eorem 2 and eorem 3.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.