Let $\mathbf{ X}_1,...,\mathbf{ X}_n$ be a random sample from a $p$-dimensional population distribution. Assume that $c_1n^{ \alpha}\leq p\leq c_2n^{ \alpha}$ 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.
Categories: math.PR