Likelihood Ratio Tests for High-Dimensional Normal Distributions

type="main" xml:id="sjos12147-abs-0001"> 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 presented afterwards.

[1]  Yumou Qiu,et al.  Test for Bandedness of High Dimensional Covariance Matrices with Bandwidth Estimation , 2012, 1208.3321.

[2]  M. Bartlett,et al.  A note on the multiplying factors for various chi square approximations , 1954 .

[3]  V. I. Serdobolskii.  Multivariate Statistical Analysis: A High-Dimensional Approach , 2000 .

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

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

[6]  T. Cai,et al.  Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings , 2013 .

[7]  S. S. Wilks.  CERTAIN GENERALIZATIONS IN THE ANALYSIS OF VARIANCE , 1932 .

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

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

[10]  S. S. Wilks.  The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses , 1938 .

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

[12]  Martin J. Wainwright,et al.  A More Powerful Two-Sample Test in High Dimensions using Random Projection , 2011, NIPS.

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

[14]  A. V. D. Vaart,et al.  Asymptotic Statistics: Frontmatter , 1998 .

[15]  Marcelo J. Moreira,et al.  Asymptotic power of sphericity tests for high-dimensional data , 2013, 1306.4867.

[16]  L. Ahlfors.  Complex Analysis , 1979 .

[17]  G. Box,et al.  A general distribution theory for a class of likelihood criteria. , 1949, Biometrika.

[18]  J. Mauchly.  Significance Test for Sphericity of a Normal $n$-Variate Distribution , 1940 .

[19]  Jun Yu Li,et al.  Two Sample Tests for High Dimensional Covariance Matrices , 2012, 1206.0917.

[20]  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..

[21]  Nariaki Sugiura,et al.  Unbiasedness of Some Test Criteria for the Equality of One or Two Covariance Matrices , 1968 .

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

[23]  D. F. Morrison,et al.  Multivariate Statistical Methods , 1968 .

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

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

[26]  T. Gamelin.  Complex Analysis , 2001 .

[27]  Michael D. Perlman,et al.  Unbiasedness of the Likelihood Ratio Tests for Equality of Several Covariance Matrices and Equality of Several Multivariate Normal Populations , 1980 .

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

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

[30]  Han Xiao,et al.  Asymptotic theory for maximum deviations of sample covariance matrix estimates , 2013 .

[31]  S. S. Wilks.  On the Independence of k Sets of Normally Distributed Statistical Variables , 1935 .