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

Abstract

Let X₁, …, X_n be a random sample from a p-dimensional population distribution. Assume that c₁n^α≤p≤c₂n^α for some positive constants c₁, c₂ 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 $O((\log n)^{5/2}/\sqrt{n})$. 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.