The asymptotic distributions of the largest entries of sample correlation matrices

Abstract

Let X n = (x ij) be an n by p data matrix, where the n rows form a random sample of size n from a certain p-dimensional population distribution. Let R n = (ρ ij) be the p × p sample correlation matrix of X n; that is, the entry ρ ij is the usual Pearson's correlation coefficient between the ith column of X n and jth column of X n. For contemporary data both n and p are large. When the population is a multivariate normal we study the test that H 0 : the p variates of the population are uncorrelated. A test statistic is chosen as L n = max i ≠ j |ρ ij|. The asymptotic distribution of L n is derived by using the Chen-Stein Poisson approximation method. Similar results for the non-Gaussian case are also derived. © Institute of Mathematical Statistics, 2004.