A study of two high-dimensional likelihood ratio tests under alternative hypotheses

Let Np(μ,Σ) be a p-dimensional normal distribution. Testing Σ equal to a given matrix or (μ,Σ) equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension p is fixed, it is known that the LRT statistics go to χ2-distributions. When p is large, simulation shows that the approximations are far from accurate. For the two LRT statistics, in the high-dimensional cases, we obtain their central limit theorems under a big class of alternative hypotheses. In particular, the alternative hypotheses are not local ones. We do not need the assumption that p and n are proportional to each other. The condition n − 1 > p →∞ suffices in our results.

[1]  James R. Schott,et al.  Testing for complete independence in high dimensions , 2005 .

[2]  Song-xi Chen,et al.  Tests for High-Dimensional Covariance Matrices , 2010, Random Matrices: Theory and Applications.

[3]  James R. Schott,et al.  A test for the equality of covariance matrices when the dimension is large relative to the sample sizes , 2007, Comput. Stat. Data Anal..

[4]  James R. Schott,et al.  Some tests for the equality of covariance matrices , 2001 .

[5]  Alexei Onatski,et al.  Signal detection in high dimension: The multispiked case , 2012, 1210.5663.

[6]  T. J. Page.  Multivariate Statistics: A Vector Space Approach , 1984 .

[7]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

[8]  Shurong Zheng,et al.  Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing , 2014, 1404.6633.

[9]  P. Spreij.  Probability and Measure , 1996 .

[10]  Tiefeng Jiang,et al.  Likelihood ratio tests for covariance matrices of high-dimensional normal distributions , 2012 .

[11]  T. Cai,et al.  Optimal hypothesis testing for high dimensional covariance matrices , 2012, 1205.4219.

[12]  Fan Yang,et al.  Likelihood Ratio Tests for High‐Dimensional Normal Distributions , 2013, 1306.0254.

[13]  M. Srivastava.  Some Tests Concerning the Covariance Matrix in High Dimensional Data , 2005 .

[14]  Olivier Ledoit,et al.  Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size , 2002 .

[15]  Z. Bai,et al.  Corrections to LRT on large-dimensional covariance matrix by RMT , 2009, 0902.0552.

[16]  Yongcheng Qi,et al.  Likelihood Ratio Tests for High-Dimensional Normal Distributions , 2015 .

[17]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.