Strong Uniform Convergence Rates of Wavelet Density Estimators with Size-Biased Data

Abstract

This paper considers the strong uniform convergence of multivariate density estimators in Besov space based on size-biased data. We provide convergence rates of wavelet estimators when the parametric is known or unknown, respectively. It turns out that the convergence rates coincide with that of Giné and Nickl’s (Uniform Limit Theorems for Wavelet Density Estimators, Ann. Probab., 37(4), 1605-1646, 2009), when the dimension , , and .

1. Introduction

Let be independent and identically distributed () continuous random variables defined on a probability space with the common density functionwhere denotes a known positive function and stands for an unknown density function of the unobserved continuous random variable and . In this setup and mean the target density and weighted density function, respectively, and the resulting data are size-biased data. Then we want to estimate the unknown density function from a sequence of biased data .

Wavelet methods are of interest in nonparametric statistics thanks to their ability to estimate efficiently a wide variety of unknown functions, especially for those with discontinuities or sharp spikes. Hence, wavelet methods have been widely used for this density estimation model (1). Ramírez and Vidakovic [1] propose a linear wavelet estimator and show it to be consistent. Shirazi and Doosti [2] expand their work to multivariate case. Chesneau, Dewan, and Doosti [3] extend the independence to both positively and negatively associated cases. They show a convergence rate for mean integrated squared error (MISE). An upper bound of wavelet estimation on risk in negatively associated case is given by Liu and Xu [4]. Kou and Guo [5] discuss the MISE of wavelet estimators in strong mixing case. For the strong convergence of density estimation, Masry [6] studies the strong convergence rates over a compact subset in Besov space , when (the model (1) reduces to the classical density estimation) and the sample is strong mixing. Recently, Giné and Nickl [7] investigate the same problem by wavelet method and obtain the optimal strong convergence rates in Besov space , when the data is . To our knowledge, there does not exist research on the strong uniform convergence for the model (1).

The aim of this paper is to discuss the strong uniform convergence rates of wavelet estimators in Besov space based on size-biased data. First of all, we construct a linear wavelet estimator when the parametric is known and give its convergence rate. However, people always do not know in many practical applications. For this reason, an estimator of is given. Then we develop a new linear wavelet estimator in which the parametric is replaced by . Finally, we establish the convergence rate of estimator .

2. Wavelets and Besov Spaces

As a central notion in wavelet analysis, Multiresolution Analysis (MRA, [8]) plays an important role for constructing a wavelet basis, which means a sequence of closed subspaces of the square integrable function space satisfying the following properties:

(i) , . Here and after, denotes the integer set and ;

(ii) . This means the space being dense in ;

(iii) if and only if for each ;

(iv) There exists a scaling function such that forms an orthonormal basis of .

When , there is a simple way to define an orthonormal wavelet basis. Examples include the Daubechies wavelets with compact supports. For , the tensor product method gives an MRA of from one-dimensional MRA. In fact, with a scaling function of tensor products, we find wavelet functions such that, for each , the decomposition holds in sense, where , , and

Let be the orthogonal projection operator from onto the space with the orthonormal basis . Then for ,

If a scaling function satisfies Condition , i.e., then the function (so that for ) and converges absolutely almost everywhere. It can be shown that, for ,holds almost everywhere on [9]. In this paper, we also need another concept, which is a little stronger than Condition .

A function is said to satisfy Condition (), if there exists a bounded and radical nonincreasing function such that Condition is not very restrictive. Examples include bounded and compactly supported measurable functions. Daubechies scaling functions satisfy Condition .

A wavelet basis can be used to characterize Besov spaces. The next lemma provides equivalent definitions for those spaces, for which we need one more notation: a scaling function is called -regular, if and for each and each multi-index with .

Lemma 1 ([8]). Let be -regular, () be the corresponding wavelets and . If , , , and , then the following assertions are equivalent:
(1) ;
(2) ;
(3)

The Besov norm of can be defined by with and .

We also need the following classical inequality in the proof of our theorems.

Bernstein’s inequality. Let be independent random variables such that , , and . Then for each ,

3. Estimation with Known

In this paper, we require supp in the model (1). This is similar to Chesneau, Dewan, and Doosti [3], Liu and Xu [4], and Kou and Guo [5]. We choose -dimensional scaling function with being the one-dimensional Daubechies scaling function. Then is -regular when gets large enough. Note that has compact support and the corresponding wavelet has compact support . Then for with supp and ,where , , and

A linear wavelet estimator is defined bywhereIt follows from (1) thatThis means is an unbiased estimate of . The following notations are needed to state our theorems. denotes for some constant ; means ; stands for both and .

Theorem 2. Consider the problem (1) with . Let ) and . Then the linear wavelet estimator defined in (12) with satisfies

Remark 3. When , our model reduces to the classical nonparametric density estimation. Then our result is same as the convergence rate in Masry [6]. On the other hand, we find that with and . This coincides with the convergence rate in Theorem 3 of Giné and Nickl [7].

Proof. It is easy to see thatBy , . Then it follows from (14) and Lemma 1 () thatThis with the choice leads toTo estimate the other term of (17), by splitting the interval equally into ( standing for the smallest integer greater than or equal to ) subintervals, one receives sub-cubes of . Clearly, the side length of satisfies that Note thatThen with the center point of ,where By the definition of ,Since the properties of imply . On the other hand, the Daubechies function satisfies Lipschitz condition () for larger . Then for , Hence, for any , Combining this with (25) and , one finds that Recalling that and , thenBy , . Furthermore, it follows from the proof of (29) thatThe main work for the proof of Theorem 2 is to estimate Set and is constant which will be chosen later. Then note thatAccording to the definition of , one concludesDenote for . Then are , . By and Condition , andThis with Bernstein’s inequality (Härdle et al., 1998) and leads toIt follows from (32), (36), , and the definition of thatObviously, there exists sufficiently large such that . Then andHence,thanks to Borel-Cantelli lemma. This with (23), (29), and (30) showsCombining this with (17) and (19), one knows that

A careful observation of (12) shows the construction of strictly depends on , which needs known. However, the parametric is always unknown in many practical applications. So we will deal with the unknown case in the following section.

4. Estimation with Unknown

In this section, we provide a strong convergence rate of wavelet estimator for the model (1) with unknown parametric . A first step is to give an estimator of from the given data . Similar to Chesneau, Dewan, and Doosti [3] and Liu and Xu [4], we introduceBy (1),Now, we define a practical linear wavelet estimatorwith

Theorem 4 investigates the strong uniform convergence rate of practical wavelet estimator .

Theorem 4. Consider the problem (1) with . Let ) and . Then the linear wavelet estimator defined in (44) with satisfies

Remark 5. Note that the convergence rate of wavelet estimator in Theorem 4 remains same as that of in Theorem 2. However, the estimator does not depend on the parametric , which means it is more practical.

Proof. By the definition of and , Then one observes thatwhere It follows from (19) thatAccording to and the definition of in (42), one gets and Then it is easy to see from (40) thatFinally, one estimate . Since , . On the other hand, thanks to Condition and Lemma 3.3 in Liu and Xu [10]. Hence,Take . Then is and (see (43)). By , and . Similar to the arguments of (39),This with (48), (50), and (52) shows

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 paper is supported by Guangxi Natural Science Foundation (Nos. 2017GXNSFAA198194 and 2018GXNSFBA281076), Guangxi Science and Technology Project (Nos. AD18281058 and AD18281019), the Guangxi Young Teachers Basic Ability Improvement Project (Nos. 2018KY0212 and 2019KY0218), and Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation.