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

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

2009-01-16

arxiv.org/abs/0901.2468v1

Annals of Applied Probability 2008, Vol. 18, No. 6, 2337-2366

Mathematics

Probability

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

Scientific paper

10.1214/08-AAP527

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.

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