The asymptotic distribution and Berry--Esseen bound of a new test for independence in high dimension with an application to stochastic optimization

Abstract

Let ${\mathbf{X}}_1,...,{\mathbf{X}}_n$ be a random sample from a $p$-dimensional population distribution. Assume that $c_1{n}^{\alpha }\le p\le c_2{n}^{\alpha }$ for some positive constants $c_1,c_2$ and $\alpha$. In this paper we introduce a new statistic for testing independence of the $p$-variates of the population and prove that the limiting distribution is the extreme distribution of type I with a rate of convergence $O((\log n{)}^{5/2}/\sqrt{n})$. This is much faster than $O(1/\log n)$, a typical convergence rate for this type of extreme distribution. A simulation study and application to stochastic optimization are discussed.

Published in at http://dx.doi.org/10.1214/08-AAP527 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)