Least Favorable Direction Test for Multivariate Analysis of Variance in High Dimension

This paper considers the problem of multivariate analysis of variance for normal samples in the high dimension medium sample size setting. When the sample dimension is larger than the sample size, the classical likelihood ratio test is not defined since the likelihood function is unbounded. Based on the unboundedness of the likelihood function, we propose a new test called the least favorable direction test. The asymptotic distributions of the test statistic are derived under both nonspiked and spiked covariances. The local asymptotic power function of the test is also given. The asymptotic power function and simulations show that the proposed test is particularly powerful under spiked covariance.

[1]  Charles R. Johnson,et al.  Matrix Analysis, 2nd Ed , 2012 .

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

[3]  James R. Schott,et al.  Some high-dimensional tests for a one-way MANOVA , 2007 .

[4]  M. Aoshima,et al.  Two-Sample Tests for High-Dimensional Linear Regression with an Application to Detecting Interactions. , 2016, Statistica Sinica.

[5]  Xingzhong Xu,et al.  A feasible high dimensional randomization test for the mean vector , 2019, Journal of Statistical Planning and Inference.

[6]  Daojiang He,et al.  A test for the k sample Behrens–Fisher problem in high dimensional data , 2017, Journal of Statistical Planning and Inference.

[7]  Makoto Aoshima,et al.  PCA consistency for the power spiked model in high-dimensional settings , 2013, J. Multivar. Anal..

[8]  Tatsuya Kubokawa,et al.  Tests for multivariate analysis of variance in high dimension under non-normality , 2013, J. Multivar. Anal..

[9]  Hansheng Wang,et al.  A High Dimensional Two-Sample Test Under a Low Dimensional Factor Structure , 2015, J. Multivar. Anal..

[10]  Lixing Zhu,et al.  TWO-SAMPLE BEHRENS-FISHER PROBLEM FOR HIGH-DIMENSIONAL DATA , 2015 .

[11]  B. Nadler.  Finite sample approximation results for principal component analysis: a matrix perturbation approach , 2009, 0901.3245.

[12]  T. Cai,et al.  Optimal estimation and rank detection for sparse spiked covariance matrices , 2013, Probability theory and related fields.

[13]  Makoto Aoshima,et al.  Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations , 2012, J. Multivar. Anal..

[14]  R. Ivry,et al.  Ipsilateral motor cortex activity during unimanual hand movements relates to task complexity. , 2005, Journal of neurophysiology.

[15]  Jianqing Fan,et al.  Large covariance estimation by thresholding principal orthogonal complements , 2011, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[16]  Song-xi Chen,et al.  A two-sample test for high-dimensional data with applications to gene-set testing , 2010, 1002.4547.

[17]  Jianqing Fan,et al.  Asymptotics of empirical eigenstructure for high dimensional spiked covariance. , 2017, Annals of statistics.

[18]  Rui Wang,et al.  On two-sample mean tests under spiked covariances , 2018, J. Multivar. Anal..

[19]  Z. Bai,et al.  EFFECT OF HIGH DIMENSION: BY AN EXAMPLE OF A TWO SAMPLE PROBLEM , 1999 .

[20]  Muni S. Srivastava,et al.  Asymptotic distributions of some test criteria for the mean vector with fewer observations than the dimension , 2013, J. Multivar. Anal..

[21]  P. Massart,et al.  Adaptive estimation of a quadratic functional by model selection , 2000 .

[22]  Dan Shen,et al.  A General Framework for Consistency of Principal Component Analysis , 2012, J. Mach. Learn. Res..

[23]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

[24]  James J. Chen,et al.  Multivariate analysis of variance test for gene set analysis , 2009, Bioinform..

[25]  Takayuki Yamada,et al.  Testing homogeneity of mean vectors under heteroscedasticity in high-dimension , 2015, J. Multivar. Anal..

[26]  Vladimir Koltchinskii,et al.  Asymptotics and Concentration Bounds for Bilinear Forms of Spectral Projectors of Sample Covariance , 2014, 1408.4643.

[27]  S. N. Roy.  On a Heuristic Method of Test Construction and its use in Multivariate Analysis , 1953 .

[28]  T. Tony Cai,et al.  High-dimensional sparse MANOVA , 2014, J. Multivar. Anal..

[29]  M. Srivastava.  Multivariate Theory for Analyzing High Dimensional Data , 2007 .

[30]  Haipeng Shen,et al.  A survey of high dimension low sample size asymptotics , 2018, Australian & New Zealand journal of statistics.

[31]  J. Marron,et al.  The maximal data piling direction for discrimination , 2010 .