Determinant of sample correlation matrix with application

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Abstract

Let x 1 ,...,x n be independent random vectors of a common p-dimensional normal distribution with population correlation matrix R n . The sample correlation matrix Ř n = (ř ij ) p × p is generated from x 1 ,...,x n such that ř ij is the Pearson correlation coefficient between the ith column and the jth column of the data matrix (x 1 ,...,x n ). The matrix Ř n is a popular object in multivariate analysis and it has many connections to other problems. We derive a central limit theorem (CLT) for the logarithm of the determinant of Ř n for a big class of R n . The expressions of mean and the variance in the CLT are not obvious, and they are not known before. In particular, the CLT holds if p/n has a nonzero limit and the smallest eigenvalue of R n is larger than 1/2. Besides, a formula of the moments of |Ř n | and a new method of showing weak convergence are introduced. We apply the CLT to a high-dimensional statistical test.

Original languageEnglish (US)
Pages (from-to)1356-1397
Number of pages42
JournalAnnals of Applied Probability
Volume29
Issue number3
DOIs
StatePublished - Jan 1 2019

Bibliographical note

Funding Information:
Received October 2016; revised August 2017. 1Supported in part by NSF Grants DMS-12-09166 and DMS-14-06279. MSC2010 subject classifications. 60B20, 60F05. Key words and phrases. Central limit theorem, sample correlation matrix, smallest eigenvalue, multivariate normal distribution, moment generating function.

Funding Information:
1Supported in part by NSF Grants DMS-12-09166 and DMS-14-06279. MSC2010 subject classifications. 60B20, 60F05.

Publisher Copyright:
© Institute of Mathematical Statistics, 2019

Keywords

  • Central limit theorem
  • Moment generating function
  • Multivariate normal distribution
  • Sample correlation matrix
  • Smallest eigenvalue

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