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 H₀: 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.