The asymptotic distributions of the largest entries of sample correlation matrices

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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 ≠ jij|. 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.

Original languageEnglish (US)
Pages (from-to)865-880
Number of pages16
JournalAnnals of Applied Probability
Volume14
Issue number2
DOIs
StatePublished - May 2004

Keywords

  • Chen-Stein method
  • Maxima
  • Moderate deviations
  • Sample correlation matrices

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