# A Nonuniform Bound to an Independent Test in High Dimensional Data Analysis via Stein’s Method

Authors: Nahathai Rerkruthairat
Journal: Journal of Probability and Statistics
 (2019)

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## Abstract

The Berry-Esseen bound for the random variable based on the sum of squared sample correlation coefficients and used to test the complete independence in high diemensions is shown by Stein’s method. Although the Berry-Esseen bound can be applied to all real numbers inR, a nonuniform bound at a real number z usually provides a sharper bound if z is fixed. In this paper, we present the first version of a nonuniform bound on a normal approximation for this random variable with an optimal rate of 1/0.5+|z|·O1/m by using Stein’s method.


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## Body

## 1. Introduction

Frequently, the relations between many variables from an experiment or data are analyzed. For example, in financial analysis, the performance of various banking sectors are investigated or it is interesting to know how land use affects soil chemical properties in geology. For the traditional procedures in multivariate analysis based on asymptotic theory, the number of variables, say,m, is fixed and the size of a sample from population, say, n, tends to infinity. These inference procedures, however, may not be suitable when m>n. Many recent articles are focused on the assumption that m>n, or when both m and n go to infinity such as Schott [1, 2], Srivastava [3], Chen, Zhang and Zhong [4], Srivastava, Kollo, and Rosen [5], and Jiang and Yang [6].LetX1,…,Xn be a random sample of size n from the population. We attend a test for the independence of m variables illustrated by a random vector Xi=(X1i,…,Xmi)′ with a covariance matrix Σ. Testing for complete independence in the population having a multivariate normal distribution is identical to testing Σ=Im. In 2005, Schott [1] introduced a simple test procedure based on the sample correlation when m and n approach infinity. Let R=(rij),1≤i,j≤m be the sample correlation matrix where(1)rij=∑k=1nXik-X¯iXjk-X¯j∑k=1nXik-X¯i2∑k=1nXjk-X¯j2,Xi=(X1i,…,Xmi)′ and X¯i=1/n∑k=1nXik. Moreover, Schott defined(2)tn,m=∑i=2m∑j=1i-1rij2andWn,m=cn,mtn,m-mm-12n-1,where cn,m=nn+2/m(m-1)(n-1) and verified the central limit theorem(3)Wn,m→dN0,1when m/n∈(0,∞) and used Wn,m to test the complete independence of the variables.LetF and G be distribution functions. Recall that a constant C satisfying the condition(4)supx∈RFx-Gx≤C is called a uniform bound and a constant C(x0) satisfying the conditon(5)Fx0-Gx0≤Cx0is called a nonuniform bound. An advantage of a uniform bound is that it can be applied to all of z in R. Nevertheless, if we have a real number x0, a nonuniform bound at x0 is alway sharper than a uniform bound.When the central limit theorem is derived, it is usual to investigate a uniform bound, a nonuniform bound, or the rate of convergence in such result. Stein’s method [7] is a powerful technique for this estimating. There are many works in many fields of mathematics that applied this method. For example, a uniform and a nonuniform bound on a normal approximation of randomized orthogonal array sampling design were obtained by using the exchangeable pair technique of Stein’s method (see [8–12]), and Poisson and normal limit laws for fringe subtrees were proved by using the coupling technique of Stein’s method (see [13]).As an application of Stein’s method for a normal approximation, Chen and Shao [14] established the first version of the Berry-Esseen bound for Wn,m under the condition Xij,1≤i≤m,1≤j≤n are i.i.d. random variables and m=O(n). If E(X1124)<∞, their result is as follows:(6)supz∈RPWm,n<z-Φz=O1mwhere Φ is the standard normal distribution.In this present work, we investigate the first version of a nonuniform bound forWn,m with an optimal rate of 1/0.5+z·O1/m, under the same conditions as the previous article [14], by using Stein’s method.


## 2. Auxiliary Results

In this section, we give some lemmas which will be useful for the next section. Throughout this article, the constantC is an absolute constant and it may be different in each situation. To find a nonuniform bound of Wn,m, we truncate rij with the condition 0.5+z. Let z>0 and define(7)Y^=cn,mt^n,m-mm-12n-1where (8)t^n,m=∑i=2m∑j=1i-1r^ij,z2r^ij,z=rijIrij2≤0.5+z,rij,z=rijIrij2>0.5+zand I is the indicator function.Clearly,E(r^ij,zp)≤Erijp when p∈N. For each i∈N,1≤i≤n, let Xi∗ be an independent copy of Xi and let I be a random index uniformly distributed over 1,2,…,m where I is independent of Xi∗,Xi,1≤i≤n. Define(9)W∗=cn,mtn,m∗-mm-12n-1andY^∗=cn,mt^n,m∗-mm-12n-1where (10)t^n,m∗=t^n,m-∑j=1j≠Imr^Ij,z2+∑j=1j≠Imr^I∗j,z2r^i∗j,z=ri∗jIri∗j2≤0.5+z,ri∗j,z=ri∗jIri∗j2>0.5+z,and(11)ri∗j=∑k=1nXik∗-X¯i∗Xjk-X¯j∑k=1nXik∗-X¯i∗2∑k=1nXjk-X¯j2.It is easy to see that (t^n,m,t^n,m∗) is an exchangeable pair and so is (Y^,Y^∗). Let Xn=X1,…,Xn and let ui=(ui1,…,uin)′ where uik=Xik-X¯i/∑l=1n(Xil-X¯i)2. In 2012, Chen and Shao [14] showed that(12)Erij2=1n-1,Erij4=3n2+O1n3ifEX116<∞,Erij8=O1n4ifEX1112<∞,andErij16=O1n8ifEX1124<∞. Denote p^ij,z=P(rij2≤0.5+z)=P(ri∗j2≤0.5+z). Recall that, for i≠j,(13)Erij2=Eui′ujuj′ui=EEui′ujuj′ui∣ui=Eui′1n-1In-1nn-11nui=1n-1 (see [14], pp.20). Then, for i≠j,(14)Er^ij,z2=Erij2Irij2≤0.5+z=EEui′ujuj′ui∣uiIrij2≤0.5+z=Eui′1n-1In-1nn-11nuiIrij2≤0.5+z=1n-1Prij2≤0.5+z=p^ij,zn-1where In is the n×n identity matrix and 1n is the n×n matrix with all entries 1. Similarly, if j1≠j2,(15)Er^ij1,z2r^ij2,z2=p^ij1,z·p^ij2,zn-12.Lemma 1.
IfE(X116)<∞, then(16)E1-m4EY^∗-Y^2∣Y^2=O1m.Proof.
Defineui∗=(ui1∗,ui2∗,…,uin∗)′ where uik∗=Xik∗-X¯i∗/∑l=1n(Xil∗-X¯i∗)2. Similar to (14), we have E(r^i∗j,z2)=p^ij,z/n-1. By the same idea as (2.32) in [14],(17)1m∑i=1mE∑j=1j≠imr^ij,z2-r^i∗j,z22∣Xn=1m∑i=1mE∑j=1j≠imr^ij,z2-p^ij,zn-1-∑j=1j≠imr^i∗j,z2-p^ij,zn-12∣Xn=1m∑i=1m∑j=1j≠imr^ij,z2-p^ij,zn-12+1m∑i=1mE∑j=1j≠imr^ij,z2-p^i∗j,zn-12∣Xn.Then (18)E1-m4EY^∗-Y^2∣Y^2=cn,m4m216E1m∑i=1m∑j=1j≠imr^ij,z2-p^ij,zn-12-2m-1n-1n2n+2+1m∑i=1mE∑j=1j≠imr^i∗j,z2-p^ij,zn-12∣Xn-2m-1n-1n2n+22≤cn,m4m216EJ1+J22+O1n2≤cn,m4m2CEJ12+EJ22+O1n2 where (19)J1=1m∑i=1m∑j=1j≠imr^ij,z2-p^ij,zn-12-2m-1n2and(20)J2=1m∑i=1mE∑j=1j≠imr^i∗j,z2-p^ij,zn-12∣Xn-2m-1n2.From (14) and (15), we have (21)1m∑i=1mE∑j=1j≠imr^ij,z2-p^ij,zn-12=1m∑i=1mE∑j=1j≠imr^ij,z2-p^ij,zn-12+∑j1=1j1≠im∑j2=1j2≠ij1≠j1mr^ij1,z2-p^ij1,zn-1r^ij2,z2-p^ij2,zn-1=1m∑i=1m∑j=1j≠imEr^ij,z4-2n-1∑j=1j≠imp^ij,zEr^ij,z2+1n-12∑j=1j≠imp^ij,z2+∑j1=1j1≠im∑j2=1j2≠ij2≠j1mEr^ij1,z2r^ij2,z2-1n-1∑j1=1j1≠im∑j2=1j2≠ij2≠j1mp^ij2,zEr^ij1,z2-1n-1∑j1=1j1≠im∑j2=1j2≠ij2≠j1mp^ij1,zEr^ij2,z2+1n-12∑j1=1j1≠im∑j2=1j2≠ij2≠j1mp^ij1,zp^ij2,z=1m∑i=1m∑j=1j≠imEr^ij,z4-1n-12∑j=1j≠imp^ij,z2=m-1Er^1j,z4-m-1p^1j,z2n-12.This implies that(22)EJ12=E1m∑i=1m∑j=1j≠imr^ij,z2-p^ij,zn-12-2m-1n22=1m2∑i=1mE∑j=1j≠imr^ij,z2-p^ij,zn-14+1m2∑i1=1m∑i2=1i2≠i1mE∑j=1j≠i1mr^i1j,z2-p^i1j,zn-12∑l=1l≠i2mr^i2l,z2-p^i2l,zn-12-4m-1n2m-1Er^1j,z4-m-1p^1j,z2n-12+4m-12n4=1m2∑i=1m∑j=1j≠imEr^ij,z2-p^ij,zn-14+4m2∑i=1m∑j1=1j1≠im∑j2=1j2≠ij2≠j1mEr^ij1,z2-p^ij1,zn-13r^ij2,z2-p^ij2,zn-1+3m2∑i=1m∑j1=1j1≠im∑j2=1j2≠ij2≠j1mEr^ij1,z2-p^ij1,zn-12r^ij2,z2-p^ij2,zn-12+6m2∑i=1m∑j1=1j1≠im∑j2=1j2≠ij2≠j1m∑j3=1j3≠ij3≠j1j3≠j2mEr^ij1,z2-p^ij1,zn-12r^ij2,z2-p^ij2,zn-1r^ij3,z2-p^ij3,zn-1+1m2∑i=1m∑j1=1j1≠im∑j2=1j2≠ij2≠j1m∑j3=1j3≠ij3≠j1j3≠j2m∑j4=1j4≠ij4≠j1j4≠j2j4≠j3mEr^ij1,z2-p^ij1,zn-1r^ij2,z2-p^ij2,zn-1r^ij3,z2-p^ij3,zn-1r^ij4,z2-p^ij4,zn-1+1m2∑i1=1m∑i2=1i2≠i1mEr^i1i2,z2-p^i1i2,zn-14+1m2∑i1=1m∑i2=1i2≠i1m∑j1=1m∑j2=1mIj1≠i2orj2≠i1Er^i1j1,z2-p^i1j1,zn-12r^i2j2,z2-p^i2j2,zn-12+4m2∑i1=1m∑i2=1i2≠i1m∑j=1j≠i1j≠i2mEr^i1i2,z2-p^i1i2,zn-12r^i1j,z2-p^i1j,zn-1r^i2j,z2-p^i2j,zn-1+Er^i1i2,z2-p^i1i2,zn-1r^i1j,z2-p^i1j,zn-12r^i2j,z2-p^i2j,zn-1-4m-12n2Er^1j,z4+4m-12p^1j,z2n2n-12+4m-12n4. Note that (23)
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. Hence, (24)EJ12=O1n4+0+O1n4+0+0+O1n4+m·O1n4+O1n3+O1n3=O1n3. Similarly, EJ22≤O(1/n3). Therefore, the proof is complete.Lemma 2.
Letg:R→R be a measurable function. Then(25)EY^gY^=m4EY^∗-Y^gY^∗-gY^.Proof.
Similar to the proof of (2.17) in [14], it is easy to see that (26)EY^∗-Y^∣Xn=cn,mm∑i=1mE∑j=1j≠im-r^ij,z2-r^i∗j,z2∣Xn=-2mY^.Since (Y^,Y^∗) is an exchangeable pair and the fact that(27)Fy,y^∗=y,y^∗gy^∗+gy is antisymmetric [15], we have (28)0=EY^∗-Y^gY^∗+gY^=EY^∗-Y^2gY^+EY^∗-Y^gY^∗-gY^=2EgY^EY^∗-Y^∣Xn+EY^∗-Y^gY^∗-gY^=-4mEY^gY^+EY^∗-Y^gY^∗-gY^.Thus EY^g(Y^)=m/4EY^∗-Y^gY^∗-gY^.Lemma 3.
Letg:R→R be a continuous and piecewise continuously differentiable function. Then(29)EY^gY^=E∫-∞∞g′Y^+tKtdt where(30)Kt=m4Y^∗-Y^I0≤t≤Y^∗-Y^-IY^∗-Y^≤t≤0.Proof.
Note that(31)EY^∫-∞∞Ktdt=m4EY^∗-Y^∣Y^.and (32)m4EY^∗-Y^gY^∗-gY^=m4EY^∗-Y^∫0Y~-Y^g′Y^+tdt=E∫-∞∞g′Y^+tKtdt.By Lemma 2, this lemma is proved.Lemma 4.
(1)
IfEX116<∞, then EY^∗-Y^2=O(1/m). (2)
IfEX1112<∞, then EY^∗-Y^4=O(1/m2). (3)
IfEX1112<∞, then EY^∗-Y^3=O(1/m3/2). (4)
IfEX1124<∞, then EY^∗-Y^6=O(1/m3).Proof.
Since(a-b)2≤2(a2+b2) for all a,b∈R, (33)EY^∗-Y^2=cn,m2m∑i=1mE∑j=1j≠imr^ij,z2-r^i∗j,z22=cn,m2m∑i=1mE∑j=1j≠imr^ij,z2-pij,zn-1-∑j=1j≠imr^i∗j,z2-pij,zn-12≤C·cn,m2m∑i=1mE∑j=1j≠imr^ij,z2-pij,zn-12=C·cn,m2m∑i=1m∑j=1j≠imEr^ij,z2-pij,zn-12+2∑j1=1j1≠im∑j2=1j2≠ij2≠j1mEr^ij1,z2-pij1,zn-1r^ij2,z2-pij2,zn-1.From (12) and (14), we have EY^∗-Y^2=O(1/m).
Similarly, we can show that(34)EY^∗-Y^4≤C·cn,m4m∑i=1mE∑j=1j≠imr^ij,z2-pij,zn-14.By using the same argument of (2.33) in [14], (35)E∑j=1j≠imr^ij,z2-pij,zn-14=∑j=1j≠imEr^ij,z2-pij,zn-14+4∑j1=1j1≠im∑j2=1j2≠ij2≠j1mEr^ij1,z2-pij1,zn-13r^ij2,z2-pij2,zn-1+3∑j1=1j1≠im∑j2=1j2≠ij2≠j1mEr^ij1,z2-pij1,zn-12r^ij2,z2-pij2,zn-12+6∑j1=1j1≠im∑j2=1j2≠ij2≠j1m∑j3=1j3≠ij3≠j1j3≠j2mEr^ij1,z2-pij1,zn-12r^ij2,z2-pij2,zn-1r^ij3,z2-pij3,zn-1+∑j1=1j1≠im∑j2=1j2≠ij2≠j1m∑j3=1j3≠ij3≠j1j3≠j2m∑j4=1j4≠ij4≠j1j4≠j2j4≠j3mEr^ij1,z2-pij1,zn-1r^ij2,z2-pij2,zn-1r^ij3,z2-pij3,zn-1r^ij4,z2-pij4,zn-1=O1m2.Hence, EY^∗-Y^4=O(1/m2). This implies that EY^∗-Y^3=O(1/m3/2) by applying Ho¨lder’s inequality. Moreover, by following the same method as above, EY^∗-Y^6=O(1/m3).Lemma 5.
(1)
IfEX116<∞, then EY^2=O(1). (2)
IfEX1112<∞, then EY^4=O(1).Proof.
( 1 ) Let g(x)=x in Lemma 2, and we obtain EY^2=m/4E(Y^∗-Y^)2. By the previous lemma, we know that EY^∗-Y^2=O(1/m). Then EY^2=O(1) as desired.
( 2 ) Applying Lemma 3 with g(x)=x3, (36)EY^4=3E∫-∞∞Y^+t2Ktdt=3m4EY^2Y^∗-Y^2+m2EY^Y^∗-Y^3+m4EY^Y^∗-Y^4.Following the same technique as Neammanee, Laipaporn, and Sungkamongkol [10], we have EY^4=O(1).


## 3. Main Result

In this section, we prove the first version of a nonuniform bound forWm,n.Theorem 6.
LetXij,1≤i≤m,1≤j≤n be i.i.d. random variables, and let Φ be the standard normal distribution. Assume m=O(n). If E(X1124)<∞, then for every z∈R(37)PWm,n<z-Φz≤10.5+z·O1m.Proof.
For convenience, we writeW instead of Wm,n. To bound PW≤z-Φz, it suffices to consider z>0 since Φ(z)=1-Φ(-z) and we can apply the result to -W when z≤0. Then, from now on, we assume z>0.
In the view of (6) and the fact that (38)1=0.5+z0.5+z<10.5+zfor0<z≤0.25,we have (39)PW≤z-Φz=O1q≤10.5+zO1qfor0<z≤0.25.Then it suffices to prove Theorem 6 only in the case of z>0.25.
Letz>0.25. Thus (40)PW≤z-Φz≤PW≠Y^+PY^≤z-Φz.Note that (41)PW≠Y^=P∑i=2m∑j=1i-1Irij2>0.5+z≥1≤E∑i=2m∑j=1i-1Irij2>0.5+z≤∑i=2m∑j=1i-1Erij80.5+z4=10.5+z4·O1m2when we apply Chebyshev’s Inequality and (12) in the first and the second inequality, respectively. For the rest of this proof, we will bound PY^≤z-Φz. Let gz be Stein’s solution of Stein’s equation gz′(w)-wgz(w)=I(w≤z)-Φ(z). Applying Stein’s method, we transform the problem of bounding the distance between P(Y^≤z) and Φ(z) into the problem of bounding suitably chosen differential operators; that is, (42)PY^≤z-Φz=Egz′Y^-EY^gzY^.By Lemma 3, we have (43)PY^≤z-Φz=Egz′Y^-E∫-∞∞gz′Y^+tKtdt=Egz′Y^-E∫-∞∞gz′Y^Ktdt+E∫-∞∞gz′Y^Ktdt-E∫-∞∞gz′Y^+tKtdt≤Egz′Y^EY^1-∫-∞∞Ktdt+E∫-∞∞gz′Y^Ktdt-E∫-∞∞gz′Y^+tKtdt≤Egz′Y^1-m4EY~-Y^2∣Y^+E∫-∞∞gz′Y^-gz′Y^+tKtdt≔A1+A2(see, e.g., [8, 12] for more techniques). To bound A1, we recall that |gz′|≤1 and one can show that E|gz′(Y^)|≤C/(0.5+z)2 (see [16], pp.248, for more details). By Lemma 1, we have (44)A1=Egz′Y^1-m4EY^∗-Y^2∣Y^≤Egz′Y^21/2E1-m4EY^∗-Y^2∣Y^21/2≤C0.5+z21/2E1-m4EY^∗-Y^2∣Y^21/2=C0.5+zE1-m4EY^∗-Y^2∣Y^21/2=C0.5+z·O1m.Next, we will bound A2. By using the same argument as bounding T1 in [8], we have(45)A2≤A2,1+A2,2+A2,3 where (46)A2,1=EIY^∗-Y^<0.5+z2∫-∞∞Iz-max⁡0,t<Y^<z-min⁡0,tKtdt,A2,2=EIY^∗-Y^<0.5+z2∫-∞∞∫t0hY^+uKtdudt,A2,3=EIY^∗-Y^≥0.5+z2∫-∞∞gz′Y^-gz′Y^+tKtdt.and, for each z>0, function h:R→R is defined by(47)hw=wgzw′.For δ>0, let fδ:R→R be defined by (48)fδt=0ift<z-2δ,0.5+t+δt-z+2δifz-2δ≤t≤z+2δ,4δ0.5+t+δift>z+2δ.Then fδ is a nondecreasing function and (49)fδ′t≥0.5+z-δifz-2δ≤t≤z+2δ,0otherwise.Moreover, we can show that (50)hw≤C0.5+zifz2≤w≤z,C0.5+z2ifz2<worw>z([15], pp.23). Similar to bounding T1 in [8], we get that (51)A2,1=20.5+zE∫t≤Y^∗-Y^fY^∗-Y^′Y^+tIz-2Y^∗-Y^<Y^<z+2Y^∗-Y^Ktdt,A2,2=mC0.5+z2EY^∗-Y^3+mC0.5+zEY^∗-Y^3IY^+Y^∗-Y^≥z2,A2,3=EIY^∗-Y^≥0.5+z2∫-∞∞Ktdt.Lemma 3 implies that(52)E∫-∞∞fδ′Y^+tKtdt=EY^fδY^.Thus(53)A2,1≤20.5+zEY^fY~-Y^Y^and (54)EY^fY^∗-Y^Y^≤4EY^Y^∗-Y^1+Y^+Y^∗-Y^≤4EY^21/2EY^∗-Y^21/2+4EY^41/2EY^∗-Y^21/2+4EY^21/2EY^∗-Y^41/2.By Lemmas 4 and 5, we obtain (55)A2,1=10.5+z·O1m,A2,2≤C0.5+z2·O1m+mC0.5+zEY^∗-Y^61/2PY^+Y^∗-Y^≥z21/2≤C0.5+z2·O1m+mC0.5+z·O1m3/2EY^4+EY^∗-Y^4z41/2=10.5+z·O1m,A2,3≤mEIY^∗-Y^≥0.5+z2Y^∗-Y^2≤mPY^∗-Y^≥0.5+z21/2EY^∗-Y^41/2≤CEY^∗-Y^20.5+z21/2=10.5+z·O1m.Hence, A2=1/0.5+z·O(1/m), and, therefore, the proof is now complete.


## 4. Conclusions

This paper presents the first version of a nonuniform bound on a normal approximation for the random variables used in an independent test in high dimensional data analysis. In order to obtain an optimal rate of1/0.5+z·O1/m, the finiteness of the 24th moment is assumed.



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*Source: 8641870-2019-02-03.xml*