Likelihood Ratio Tests for High-Dimensional Normal Distributions

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In their recent work, Jiang and Yang studied six classical Likelihood Ratio Test statistics under high-dimensional setting. Assuming that a random sample of size n is observed from a p-dimensional normal population, they derive the central limit theorems (CLTs) when p and n are proportional to each other, which are different from the classical chi-square limits as n goes to infinity, while p remains fixed. In this paper, by developing a new tool, we prove that the mentioned six CLTs hold in a more applicable setting: p goes to infinity, and p can be very close to n. This is an almost sufficient and necessary condition for the CLTs. Simulations of histograms, comparisons on sizes and powers with those in the classical chi-square approximations and discussions are presentedafterwards.

Original languageEnglish (US)
Pages (from-to)988-1009
Number of pages22
JournalScandinavian Journal of Statistics
Issue number4
StatePublished - Dec 1 2015

Bibliographical note

Publisher Copyright:
© 2015 Board of the Foundation of the Scandinavian Journal of Statistics.


  • Central limit theorem
  • Covariance matrix
  • High-dimensional data
  • Hypothesis test
  • Likelihood ratio test
  • Mean vector
  • Multivariate Gamma function
  • Multivariate normal distribution


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