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

Authors: Huijun Guo; Junke Kou
Journal: Journal of Function Spaces
 (2019)

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## Abstract

This paper considers the strong uniform convergence of multivariate density estimators in Besov spaceBp,qs(Rd) based on size-biased data. We provide convergence rates of wavelet estimators when the parametric μ 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 d=1, p=q=∞, and ω(y)≡1.


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## Body

## 1. Introduction

LetY1,Y2,…,Yn be independent and identically distributed (i.i.d) continuous random variables defined on a probability space (Ω,F,P) with the common density function(1)gy=ωyfyμ,y∈Rd,where ω denotes a known positive function and f stands for an unknown density function of the unobserved continuous random variable X and μ=Eω(X)=∫Rdω(y)f(y)dy<+∞. In this setup f and g 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 f from a sequence of biased data Y1,Y2,…,Yn.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 L2 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 Lp(1≤p<+∞) 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 Bp,qs(Rd), when ω(y)≡1 (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 B∞,∞s(R), when the data is i.i.d. 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 spaceBp,qs(Rd) based on size-biased data. First of all, we construct a linear wavelet estimator f~n when the parametric μ is known and give its convergence rate. However, people always do not know μ in many practical applications. For this reason, an estimator μ^ of μ is given. Then we develop a new linear wavelet estimator f^n in which the parametric μ is replaced by μ^. Finally, we establish the convergence rate of estimator f^n.


## 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 {Vj}j∈Z of the square integrable function space L2(Rd) satisfying the following properties:(i)Vj⊆Vj+1, j∈Z. Here and after, Z denotes the integer set and N≔{n∈Z,n≥0};(ii)⋃j∈ZVj¯=L2(Rd). This means the space ⋃j∈ZVj being dense in L2(Rd);(iii)f(2·)∈Vj+1 if and only if f(·)∈Vj for each j∈Z;(iv) There exists a scaling functionφ∈L2(Rd) such that {φ(·-k),k∈Zd} forms an orthonormal basis of V0=span¯{φ(·-k)}.Whend=1, there is a simple way to define an orthonormal wavelet basis. Examples include the Daubechies wavelets with compact supports. For d≥2, the tensor product method gives an MRA {Vj} of L2(Rd) from one-dimensional MRA. In fact, with a scaling function φ of tensor products, we find M=2d-1 wavelet functions ψl(l=1,2,…,M) such that, for each f∈L2(Rd), the decomposition (2)f=∑k∈Zdαj0,kφj0,k+∑j=j0∞∑l=1M∑k∈Zdβj,klψj,klholds in L2(Rd) sense, where αj0,k=〈f,φj0,k〉, βj,kl=〈f,ψj,kl〉, and (3)φj0,ky=2j0d/2φ2j0y-k,ψj,kly=2jd/2ψl2jy-k.LetPj be the orthogonal projection operator from L2(Rd) onto the space Vj with the orthonormal basis {φj,k(·)=2jd/2φ(2j·-k),k∈Zd}. Then for f∈L2(Rd), Pjf=∑k∈Zdαj,kφj,k.If a scaling functionφ satisfies Condition (θ), i.e., (4)∑k∈Zdφy-k∈L∞Rd,then the function φ∈L(Rd)⋂L∞(Rd)(so that φ∈Lp for 1≤p≤∞) and ∑k∈Zdφ(x-k)φ(y-k)¯ converges absolutely almost everywhere. It can be shown that, for f∈Lp(Rd)(1≤p≤∞),(5)Pjfy=∑k∈Zdαj,kφj,kyholds almost everywhere on Rd [9]. In this paper, we also need another concept, which is a little stronger than Condition (θ).A functionφ is said to satisfy Condition (S), if there exists a bounded and radical nonincreasing function Φ such that (6)φy≤Φya.e.and∫RdΦydy<∞. Condition (S) is not very restrictive. Examples include bounded and compactly supported measurable functions. Daubechies scaling functions satisfy Condition (S).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 functionφ is called m-regular, if φ∈Cm(Rd) and Dαφy≤c(1+y2)-l for each l∈Z and each multi-index α∈Nd with α≤m.Lemma 1 ([8]).
Letφ be m-regular, ψl (l=1,2,…,M,M=2d-1) be the corresponding wavelets and f∈Lp(Rd). If αj,k=〈f,φj,k〉, βj,kl=〈f,ψj,kl〉, p,q∈[1,+∞], and 0<s<m, then the following assertions are equivalent:
(1)f∈Bp,qs(Rd);
(2){2jsPjf-fp}∈lq;
(3)αj0p+(2js-d/p+d/2βjp)j≥j0q<+∞.The Besov norm off can be defined by (7)fBp,qs≔αj0p+2js-d/p+d/2βjpj≥j0qwith (αj0)pp=∑k∈Zd|αj0,k|p and βjpp=∑l=1M∑k∈Zd|βj,kl|p.We also need the following classical inequality in the proof of our theorems.Bernstein’s inequality. Let Y1,…,Yn be independent random variables such that EYi=0, |Yi|≤M, and EYi2=σ2. Then for each v≥0, (8)P1n∑i=1nYi≥v≤2·exp⁡-nv22σ2+vM/3.


## 3. Estimation with Knownμ

In this paper, we require suppYi⊆[0,1]d in the model (1). This is similar to Chesneau, Dewan, and Doosti [3], Liu and Xu [4], and Kou and Guo [5]. We choose d-dimensional scaling function (9)φy=φy1,…,yd≔D2Ny1⋯D2Nydwith D2N(·) being the one-dimensional Daubechies scaling function. Then φ is m-regular (m>0) when N gets large enough. Note that D2N has compact support [0,2N-1] and the corresponding wavelet has compact support [-N+1,N]. Then for f∈L2(Rd) with supp f⊆[0,1]d and M=2d-1,(10)fy=∑k∈Λj0αj0,kφj0,ky+∑j=j0∞∑l=1M∑k∈Λjβj,klψj,kly,where Λj0={1-2N,2-2N,…,2j0}d, Λj={-N,-N+1,…,2j+N-1}d, and(11)αj0,k=∫0,1dfyφj0,kydy,βj,kl=∫0,1dfyψj,klydy.A linear wavelet estimator is defined by(12)f~ny=∑k∈Λj0α~j0,kφj0,ky,where(13)α~j0,k=μn∑i=1nφj0,kYiωYi.It follows from (1) that(14)Eμn∑i=1nφj0,kYiωYi=Eμφj0,kYiωYi=∫0,1dμφj0,kyωygydy=∫0,1dφj0,kyfydy=αj0,k.This means α~j0,k is an unbiased estimate of αj0,k. The following notations are needed to state our theorems. 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.Theorem 2.
Consider the problem (1) with ω(y)~1. Let f∈Bp,qs(Rd)(p,q∈[1,∞],s>d/p) and suppf⊆[0,1]d. Then the linear wavelet estimator f~n defined in (12) with 2j0~n/ln⁡n1/(2(s-d/p)+d) satisfies(15)supy∈0,1df~ny-fy=Oa.s.ln⁡nns-d/p/2s-d/p+d.Remark 3.
Whenω(y)≡1, 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 (16)supy∈0,1f~ny-fy=Oa.s.ln⁡nns/2s+1 with d=1 and p=q=∞. This coincides with the convergence rate in Theorem 3 of Giné and Nickl [7].Proof.
It is easy to see that(17)supy∈0,1df~ny-fy≤supy∈0,1df~ny-Ef~ny+supy∈0,1dEf~ny-fy.By s>d/p, Bp,qs(Rd)⊆B∞,∞s-d/p(Rd). Then it follows from (14) and Lemma 1 (p=∞) that(18)supy∈0,1dEf~ny-fy≤supy∈RdEf~ny-fy=Pj0f-f∞≲2-j0s-d/p.This with the choice 2j0~n/ln⁡n1/(2(s-d/p)+d) leads to(19)supy∈0,1dEf~ny-fy≲ln⁡nns-d/p/2s-d/p+d.
To estimate the other term of (17), by splitting the interval [0,1] equally into (20)Ln=n2j03d+2ln⁡n1/2 (⌈x⌉ standing for the smallest integer greater than or equal to x) subintervals, one receives Lnd sub-cubes I(l)(l=1,2,…,Lnd) of [0,1]d. Clearly, the side length ln of I(l) satisfies that (21)ln≲n2j03d+2ln⁡n-1/2.
Note that(22)supy∈0,1df~ny-Ef~ny=max1≤l≤Lndsupy∈Ilf~ny-Ef~ny.Then with the center point y(l) of I(l),(23)supy∈0,1df~ny-Ef~ny≤Q1+Q2+Q3,where (24)Q1≔max1≤l≤Lndsupy∈Ilf~ny-f~nyl,Q2≔max1≤l≤Lndf~nyl-Ef~nyl,Q3≔max1≤l≤Lndsupy∈IlEf~ny-Ef~nyl.
By the definition off~n(y),(25)Q1≤max1≤l≤Lndsupy∈Il∑k∈Λj0α~j0,kφj0,ky-φj0,kyl.Since ω(y)~1 the properties of φ imply α~j0,k≲2j0d/2. On the other hand, the Daubechies function D2N satisfies Lipschitz condition (|D2N(x)-D2N(y)|≲|x-y|) for larger N. Then for φ(y)=∏i=1dD2N(yi), (26)φx-φy≤dD2N∞d-1sup1≤i≤dD2Nxi-D2Nyi≲sup1≤i≤dxi-yi. Hence, for any y∈I(l), (27)φj0,ky-φj0,kyl≲2j0d/2sup1≤i≤d2j0yi-yil≲2j0d/2+1ln. Combining this with (25) and |Λi0|≲2j0d, one finds that (28)Q1≲2j02d+1ln. Recalling that ln≲n2j0(3d+2)/ln⁡n-1/2 and 2j0~n/ln⁡n1/(2(s-d/p)+d), then(29)Q1≲ln⁡nns-d/p/2s-d/p+d.
By|X|≤M~, |EX|≤M~. Furthermore, it follows from the proof of (29) that(30)Q3=max1≤l≤Lndsupy∈IlEf~ny-Ef~nyl=max1≤l≤Lndsupy∈IlEf~ny-f~nyl≲ln⁡nns-d/p/2s-d/p+d.
The main work for the proof of Theorem2 is to estimate (31)Q2=max1≤l≤Lndf~nyl-Ef~nyl. Set ηn≔ln⁡n/n(s-d/p)/(2(s-d/p)+d) and c∗>0 is constant which will be chosen later. Then note that(32)PQ2≥c∗ηn=Pmax1≤l≤Lndf~nyl-Ef~nyl≥c∗ηn≤∑l=1LndPf~nyl-Ef~nyl≥c∗ηn≤Lndsupy∈0,1dPf~ny-Ef~ny≥c∗ηn.
According to the definition off~n(y), one concludes(33)f~ny-Ef~ny=∑k∈Λj0α~j0,k-Eα~j0,kφj0,ky=1n∑i=1n∑k∈Λj0μφj0,kYiωYi-Eμφj0,kYiωYiφj0,ky.
Denote(34)Ziy≔∑k∈Λj0μφj0,kYiωYi-Eμφj0,kYiωYiφj0,ky for i=1,2,…,n. Then Z1,Z2,…,Zn are i.i.d, E(Zi)=0. By ω(y)~1 and Condition (θ), |Zi(y)|≲2j0d and(35)EZi2=var∑k∈Λj0μφj0,kYiωYiφj0,ky≤E∑k∈Λj0μφj0,kYiωYiφj0,ky2≲∫0,1d∑k∈Λj0φj0,kxφj0,ky2gxdx≲2j0d.This with Bernstein’s inequality (Härdle et al., 1998) and ηn=ln⁡n/n(s-d/p)/(2(s-d/p)+d) leads to(36)Pf~ny-Ef~ny≥c∗ηn=P1n∑i=1nZi≥c∗ηn≤2exp⁡-nc∗ηn222j0d+2j0dc∗ηn/3≤2exp⁡-nc∗2ln⁡n/n2s-d/p/2s-d/p+d2n/ln⁡nd/2s-d/p+d+1/3c∗n/ln⁡nd/2s-d/p+d≤2exp⁡-c∗2ln⁡n/2+1/3c∗=2n-c∗2/2+1/3c∗.
It follows from (32), (36), 2j0~n/ln⁡n1/(2(s-d/p)+d), and the definition of Ln that(37)PQ2≥c∗ηn≤Lndsupy∈0,1dPf~ny-Ef~ny≥c∗ηn≤n2j03d+2ln⁡nd/22n-c∗2/2+1/3c∗≤2nds-d/p+2d+1/2s-d/p+d-c∗2/2+1/3c∗.Obviously, there exists sufficiently large c∗>0 such that nd(s-d/p+2d+1)/(2(s-d/p)+d)-c∗2/(2+1/3c∗)≤2n-2. Then P{Q2≥c∗ηn}≤2n-2 and(38)∑n=1∞PQ2≥c∗ηn<+∞.Hence,(39)Q2=max1≤l≤Lndf~nyl-Ef~nyl=Oa.s.ln⁡nns-d/p/2s-d/p+dthanks to Borel-Cantelli lemma. This with (23), (29), and (30) shows(40)supy∈0,1df~ny-Ef~ny=Oa.s.ln⁡nns-d/p/2s-d/p+d.Combining this with (17) and (19), one knows that (41)supy∈0,1df~ny-fy=Oa.s.ln⁡nns-d/p/2s-d/p+d.A careful observation of (12) shows the construction of f~n(y) strictly depends on μ, which needs μ known. However, the parametric μ is always unknown in many practical applications. So we will deal with the unknown case in the following section.


## 4. Estimation with Unknownμ

In this section, we provide a strong convergence rate of wavelet estimator for the model (1) with unknown parametric μ. A first step is to give an estimator of μ from the given data Y1,Y2,…,Yn. Similar to Chesneau, Dewan, and Doosti [3] and Liu and Xu [4], we introduce(42)μ^n=1n∑i=1n1ωYi-1.By (1),(43)E1μ^n=E1n∑i=1n1ωYi=E1ωYi=∫0,1dgyωydy=1μ∫0,1dfydy=1μ.Now, we define a practical linear wavelet estimator(44)f^ny=∑k∈Λj0α^j0,kφj0,kywith(45)α^j0,k=μ^nn∑i=1nφj0,kYiωYi.Theorem4 investigates the strong uniform convergence rate of practical wavelet estimator f^n(y).Theorem 4.
Consider the problem (1) with ω(y)~1. Let f∈Bp,qs(Rd)(p,q∈[1,∞],s>d/p) and suppf⊆[0,1]d. Then the linear wavelet estimator f^n defined in (44) with 2j0~n/ln⁡n1/(2(s-d/p)+d) satisfies(46)supy∈0,1df^ny-fy=Oa.s.ln⁡nns-d/p/2s-d/p+d.Remark 5.
Note that the convergence rate of wavelet estimatorf^n(y) in Theorem 4 remains same as that of f~n(y) in Theorem 2. However, the estimator f^n(y) does not depend on the parametric μ, which means it is more practical.Proof.
By the definition ofα^j0,k and α~j0,k, (47)α^j0,k=μ^nμα~j0,kandf^ny=μ^nμf~ny. Then one observes that(48)supy∈0,1df^ny-fy=supy∈0,1dμ^nμf~ny-fy≤T1+T2+T3where (49)T1≔supy∈0,1dμ^nμf~ny-Ef~ny,T2≔supy∈0,1dμ^nμ-1Ef~ny,T3≔supy∈0,1dfy-Ef~ny.
It follows from (19) that(50)T3=supy∈0,1dfy-Ef~ny≲ln⁡nns-d/p/2s-d/p+d.According to ω(y)~1 and the definition of μ^n in (42), one gets |μ^n/μ|≲1 and (51)supy∈0,1dμ^nμf~ny-Ef~ny≲supy∈0,1df~ny-Ef~ny.
Then it is easy to see from (40) that(52)T1=Oa.s.ln⁡nns-d/p/2s-d/p+d.
Finally, one estimateT2. Since ω(y)~1, |μ^n|≲1. On the other hand, E|f~n(y)|≲1 thanks to Condition (S) and Lemma 3.3 in Liu and Xu [10]. Hence,(53)T2=supy∈0,1dμ^nμ-1Ef~ny≤supy∈0,1dμ^n1μ^n-1μEf~ny≲1μ^n-1μ=1n∑i=1n1ωYi-1μ.
TakeWi=1/ω(Yi)-1/μ. Then W1,W2,…,Wn is i.i.d and EWi=0 (see (43)). By ω(y)~1, Wi≲1 and E(Wi)2≲1. Similar to the arguments of (39),(54)T2=Oa.s.ln⁡nns-d/p/2s-d/p+d.This with (48), (50), and (52) shows(55)supy∈0,1df^ny-fy=Oa.s.ln⁡nns-d/p/2s-d/p+d.



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