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

Abstract

Let X 1, . . . , X n be a random sample from a p-dimensional population distribution. Assume that c 1 n α ≤ p ≤ c 2n α for some positive constants c 1, c 2 and α. 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/ √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. © Institute of Mathematical Statistics, 2008.