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.
1. Introduction and Preliminary
Assume that is a probability space. are independent and identically distributed (i.i.d) random variables. They have the same model , where is a real random variable and denotes a random noise (error). Furthermore, and are independent of each other. Let be the unknown probability density of and be the density of . So the probability density of is equivalent to the convolution of and . If degenerates to the diac functional , reduces to be noise-free. So, approximating the density by an estimator can be recognized as a deconvolution problem. A wavelet estimator means that 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 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 in satisfy the following properties:(1)(2)(3)(4)There exists a function vector , such that forms an orthogonal basis of the subspace , where and
For every , the space can be defined as an orthogonal complement of into , i.e., . Then, . There exists a function vector such that forms an orthogonal basis of the subspace , where . Moreover, is called a multiscaling function with multiplicity , and is called its corresponding multiwavelet.
So, if a function , it has the following expansion:where , and , .
Generally, assume that and are orthogonal projections from the space to and , respectively. Then,
Thus, .
And we also define the notation .
Moreover, the Fourier transform of is defined by
And the inverse transform of is denoted by
So, the Fourier transform of is defined as
In this paper, choose a multiscaling function with multiplicity satisfying the following condition:(C1) and with , . Note: denotes two variables satisfying , for some constant ; is equivalent to , and means both and . 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 satisfies . In fact, . By using integration by parts, Then, According to multiplicity multiresolution analysis (MMRA), where , , , and . Moreover, are bounded. So, are bounded, where is constructed by (seen in [7–9]). Thus,In addition, the density function of the random noise satisfies the following conditions [2]:(C2)(C3)
Under these two conditions, the random noise 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 Section 3.
Similar to the discussion in [2, 3, 5], if , the estimators can be defined as
According to equation (10), the linear multiwavelet estimator can be defined by
By deducing simply, we have
So . And we have the following conclusion.
Theorem 1. Assume that is defined in equation (10). Then, is an unbiased estimation of , i.e., and .
The detailed proof of Theorem 1 is similar to the proof of Lemma 2.2 in [1].
According to the definition of in equation (10), the estimator can be rewritten as
To simplify the above expansion, the functionis introduced. It is similar to the discussion in [2, 3, 5]. Then,
We introduce . Then,
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. For , the conditions (C1–C3) hold. Then, for every , satisfies:
Proof. Assume that , for every . Then,Sinceand ,We compute the derivativeAccording to the conditions (C1–C3) and ,Similarly,Then,So, the derivative functions and are bounded. By integration by parts,If , it is obtained thatIf , the conclusion holds thatFor every , and .
If ,Then,If ,Thus, for ,And if ,So, for every ,Hence, for every ,The conclusion is similar to Lemma 2.1 in [2]. According to the above conclusion, we have the following lemma.
Lemma 2. For , under the conditions (C1–C3), define the function . Then, for every , and satisfy the following:(1)(2)(3), where is defined in equation (2)
Proof. According to the conclusion of Lemma 1,for every . Then,where for fixed .
According to the definition of ,On the other hand, .
According to the condition C1 and integration by parts, we havethat is,So,SinceOn the other hand, implies . So the conclusion (2) holds.
For the conclusion (10), according to Lemma 1 and the above discussion in conclusion (2),Next, to prove the conclusion (11),Note that , thenSoBy the definition of in equation (10) and Fubini theorem,According to the convolution ,The final equality is due to Plancheral formula:The conclusion (11) holds.
Lemma 3 (see 2, Theorem 3.1). Assume that are symmetric functions, are i.i.d random variables and . If , , andThen, , where and .
3. Asymptotic Normality
In this section, asymptotic normality is discussed for the linear multiwavelet deconvolution estimator and the nonlinear estimator .
Theorem 2. Under the condition (C1–C3), with and , if , then the linear estimator satisfieswith and , where and are defined by equations (55) and (62).
Proof. Since , andSo and are orthogonal in . We haveAssume thatDefineAccording to the independence of and , we haveAccording to Lemma 2 and , we haveMoreover,By Markov’s inequality, ,According to the independence of ,If , then for arbitrary given , . Moreover, and . So can be denoted byLetsoIt is easy to check that are symmetric functions. It is similar to the work of Theorem A in [2] that and satisfy the condition of Lemma 3. According to Lemma 2 and Lemma 3,where .
Thus,The detail discussion is similar to the proof of Theorem A in [2]. So,Consider the nonlinear multiwavelet estimator
Theorem 3. Under the conditions (C1–C3), with and , if , then the nonlinear estimator satisfieswith , and , where and 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 model . Construct the data by the function “randn” and error data by the function “rand” in Matlab. That is, is a standard normal random variable and is a uniform random variable. So is the convolution of and , where and .
By the formula of the convolution, we have density function of random variable as follows:
In Figure 1, random data 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 and the density of data is shown by the red solid curve.
(a)
(b)
According to Theorem 1 and , 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 density of 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 of is given by the black solid line and the expectation of linear multiwavelet deconvolution density estimator 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 is given by the black solid line and the expectation of nonlinear multiwavelet density estimator is shown by the red solid line, where can be denoted by
(a)
(b)
(c)
(d)
In the second row and first column of Figure 2, nonlinear multiwavelet deconvolution density estimator is given by the black solid line and the expectation of nonlinear multiwavelet density estimator is shown by the red solid line, where can be denoted by
Moreover, asymptotic normality is identified by the Jarque–Bera test. The results of the J-B test are given for , , and in Table 1.
In Table 1, all results of normality are zero, and the original assumption of normal distribution can be accepted by the value of the Jarque–Bera test. These imply the conclusions of Theorem 2 and Theorem 3.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Planning of Philosophy and Social Sciences in Zhejiang Province of China (19NDQN340YB) and National Natural Science Foundation of China (11771100).