Journal of Probability and Statistics
Volume 2019 (2019), Article ID 8641870, 10 pages
https://doi.org/10.1155/2019/8641870

Research Article

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

Nahathai Rerkruthairat

Department of Mathematics, Faculty of Science, Srinakharinwirot University, Bangkok, 10110, Thailand

Correspondence should be addressed to Nahathai Rerkruthairat; nahathai@g.swu.ac.th

Received 7 September 2018; Revised 20 December 2018; Accepted 2 January 2019; Published 3 February 2019

Academic Editor: Alessandro De Gregorio

Copyright © 2019 Nahathai Rerkruthairat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 in , a nonuniform bound at a real number usually provides a sharper bound if 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 by using Stein’s method.

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, , is fixed and the size of a sample from population, say, , tends to infinity. These inference procedures, however, may not be suitable when . Many recent articles are focused on the assumption that , or when both and 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].

Let be a random sample of size from the population. We attend a test for the independence of variables illustrated by a random vector with a covariance matrix . Testing for complete independence in the population having a multivariate normal distribution is identical to testing . In 2005, Schott [1] introduced a simple test procedure based on the sample correlation when and approach infinity. Let be the sample correlation matrix where

and . Moreover, Schott defined

where and verified the central limit theorem

when and used to test the complete independence of the variables.

Let and be distribution functions. Recall that a constant satisfying the condition

is called a uniform bound and a constant satisfying the conditon

is called a nonuniform bound. An advantage of a uniform bound is that it can be applied to all of in . Nevertheless, if we have a real number , a nonuniform bound at 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 under the condition are i.i.d. random variables and . If , their result is as follows:

where is the standard normal distribution.

In this present work, we investigate the first version of a nonuniform bound for with an optimal rate of , 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 constant is an absolute constant and it may be different in each situation. To find a nonuniform bound of , we truncate with the condition . Let and define

where

and is the indicator function.

Clearly, when . For each , let be an independent copy of and let be a random index uniformly distributed over where is independent of . Define

where

and

It is easy to see that is an exchangeable pair and so is . Let and let where . In 2012, Chen and Shao [14] showed that

Denote . Recall that, for ,

(see [14], pp.20). Then, for ,

where is the identity matrix and is the matrix with all entries . Similarly, if ,

Lemma 1. If , then

Proof. Define where . Similar to (14), we have . By the same idea as (2.32) in [14],

Then

where

and

From (14) and (15), we have

This implies that

Note that

Hence,

Similarly, Therefore, the proof is complete.

Lemma 2. Let be a measurable function. Then

Proof. Similar to the proof of (2.17) in [14], it is easy to see that

Since is an exchangeable pair and the fact that

is antisymmetric [15], we have

Thus

Lemma 3. Let be a continuous and piecewise continuously differentiable function. Then

where

Proof. Note that

and

By Lemma 2, this lemma is proved.

Lemma 4.

(1)If , then .(2)If , then .(3)If , then .(4)If , then .

Proof. Since for all ,

From (12) and (14), we have .
Similarly, we can show that

By using the same argument of (2.33) in [14],

Hence, . This implies that by applying Hlder’s inequality. Moreover, by following the same method as above, .

Lemma 5.

(1)If , then .(2)If , then .

Proof. Let in Lemma 2, and we obtain . By the previous lemma, we know that . Then as desired.
Applying Lemma 3 with ,

Following the same technique as Neammanee, Laipaporn, and Sungkamongkol [10], we have .

3. Main Result

In this section, we prove the first version of a nonuniform bound for .

Theorem 6. Let be i.i.d. random variables, and let be the standard normal distribution. Assume . If , then for every

Proof. For convenience, we write instead of . To bound , it suffices to consider since and we can apply the result to when . Then, from now on, we assume .
In the view of (6) and the fact that

we have

Then it suffices to prove Theorem 6 only in the case of .
Let . Thus

Note that

when we apply Chebyshev’s Inequality and (12) in the first and the second inequality, respectively. For the rest of this proof, we will bound . Let be Stein’s solution of Stein’s equation . Applying Stein’s method, we transform the problem of bounding the distance between and into the problem of bounding suitably chosen differential operators; that is,

By Lemma 3, we have

(see, e.g., [8, 12] for more techniques). To bound , we recall that and one can show that (see [16], pp.248, for more details). By Lemma 1, we have

Next, we will bound . By using the same argument as bounding in [8], we have

where

and, for each , function is defined by

For , let be defined by

Then is a nondecreasing function and

Moreover, we can show that

([15], pp.23). Similar to bounding in [8], we get that

Lemma 3 implies that

Thus

and

By Lemmas 4 and 5, we obtain

Hence, , 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 of , the finiteness of the moment is assumed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

This research is supported by the SWU Endowment Fund Year 2017.

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