A Nonuniform Bound to an Independent Test in High Dimensional Data Analysis via Stein’s Method
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.
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 , Chen, Zhang and Zhong , Srivastava, Kollo, and Rosen , and Jiang and Yang .
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  introduced a simple test procedure based on the sample correlation when and approach infinity. Let be the sample correlation matrix where and . Moreover, Schott definedwhere and verified the central limit theoremwhen 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 conditonis 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  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 ).
As an application of Stein’s method for a normal approximation, Chen and Shao  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 , 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 definewhere 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 . Definewhere andIt is easy to see that is an exchangeable pair and so is . Let and let where . In 2012, Chen and Shao  showed that Denote . Recall that, for , (see , 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 ,Then where andFrom (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
Lemma 3. Let be a continuous and piecewise continuously differentiable function. Then where
Proof. Note thatand 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 , 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 , 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 , pp.248, for more details). By Lemma 1, we have Next, we will bound . By using the same argument as bounding in , we have where and, for each , function is defined byFor , let be defined by Then is a nondecreasing function and Moreover, we can show that (, pp.23). Similar to bounding in , we get that