Likelihood ratio tests for covariance matrices of high-dimensional normal distributions

Dandan Jiang, Tiefeng Jiang, Fan Yang

Research output: Contribution to journalArticlepeer-review

39 Scopus citations

Abstract

For a random sample of size n obtained from a p-variate normal population, the likelihood ratio test (LRT) for the covariance matrix equal to a given matrix is considered. By using the Selberg integral, we prove that the LRT statistic converges to a normal distribution under the assumption p/n→y∈(0, 1]. The result for y=1 is much different from the case for y∈(0, 1). Another test is studied: given two sets of random observations of sample size n 1 and n 2 from two p-variate normal distributions, we study the LRT for testing the two normal distributions having equal covariance matrices. It is shown through a corollary of the Selberg integral that the LRT statistic has an asymptotic normal distribution under the assumption p/n 1→y 1∈(0, 1] and p/n 2→y 2∈(0, 1]. The case for max{y 1, y 2}=1 is much different from the case max{y 1, y 2}<1.

Original languageEnglish (US)
Pages (from-to)2241-2256
Number of pages16
JournalJournal of Statistical Planning and Inference
Volume142
Issue number8
DOIs
StatePublished - Aug 2012

Keywords

  • Gamma function
  • High-dimensional data
  • Selberg integral
  • Testing on covariance matrices

Fingerprint Dive into the research topics of 'Likelihood ratio tests for covariance matrices of high-dimensional normal distributions'. Together they form a unique fingerprint.

Cite this