Source: Ann. Statist.
Volume 30, Number 4
This paper analyzes whether standard covariance matrix tests work when dimensionality is large, and in particular larger than sample size. In the latter case, the singularity of the sample covariance matrix makes likelihood ratio tests degenerate, but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well-defined. We study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together, with their ratio converging to a finite nonzero limit. We find that the existing test for sphericity is robust against high dimensionality, but not the test for equality of the covariance matrix to a given matrix. For the latter test, we develop a new correction to the existing test statistic that makes it robust against high dimensionality.
ALALOUF, I. S. (1978). An explicit treatment of the general linear model with singular covariance matrix. Sankhy¯a Ser. B 40 65-73.
ANDERSON, T. W. (1984). An Introduction to Multivariate Statistical Analy sis, 2nd ed. Wiley, New York.
Mathematical Reviews (MathSciNet): MR771294
ARHAROV, L. V. (1971). Limit theorems for the characteristic roots of a sample covariance matrix. Soviet Math. Dokl. 12 1206-1209.
BAI, Z. D. (1993). Convergence rate of expected spectral distributions of large random matrices. II. Sample covariance matrices. Ann. Probab. 21 649-672.
BAI, Z. D., KRISHNAIAH, P. R. and ZHAO, L. C. (1989). On rates of convergence of efficient detection criteria in signal processing with white noise. IEEE Trans. Inform. Theory 35 380-388.
Mathematical Reviews (MathSciNet): MR999652
BAI, Z. D. and SARANADASA, H. (1996). Effect of high dimension: By an example of a two sample problem. Statist. Sinica 6 311-329.
COHEN, A. and STRAWDERMAN, W. E. (1971). Unbiasedness of tests for homogeneity of variances. Ann. Math. Statist. 42 355-360.
Mathematical Reviews (MathSciNet): MR275563
DEMPSTER, A. P. (1958). A high dimensional two sample significance test. Ann. Math. Statist. 29 995-1010.
DEMPSTER, A. P. (1960). A significance test for the separation of two highly multivariate small samples. Biometrics 16 41-50.
GIRKO, V. L. (1979). The central limit theorem for random determinants. Theory Probab. Appl. 24 729-740.
Mathematical Reviews (MathSciNet): MR550529
GIRKO, V. L. (1988). Spectral Theory of Random Matrices. Nauka, Moscow (in Russian).
Mathematical Reviews (MathSciNet): MR955497
GLESER, L. J. (1966). A note on the sphericity test. Ann. Math. Statist. 37 464-467.
JOHN, S. (1971). Some optimal multivariate tests. Biometrika 58 123-127.
JOHN, S. (1972). The distribution of a statistic used for testing sphericity of normal distributions. Biometrika 59 169-173.
JONSSON, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 1-38.
LÄUTER, J. (1996). Exact t and F tests for analyzing studies with multiple endpoints. Biometrics 52 964-970.
MAR CENKO, V. A. and PASTUR, L. A. (1967). Distribution of eigenvalues for some sets of random matrices. Math. U.S.S.R. Sbornik 1 457-483.
MARSHALL, A. W. and OLKIN, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
MUIRHEAD, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
NAGAO, H. (1973). On some test criteria for covariance matrix. Ann. Statist. 1 700-709.
NARAy ANASWAMY, C. R. and RAGHAVARAO, D. (1991). Principal component analysis of large dispersion matrices. Appl. Statist. 40 309-316.
SARANADASA, H. (1993). Asy mptotic expansion of the misclassification probabilities of Dand A-criteria for discrimination from two high-dimensional populations using the theory of large-dimensional random matrices. J. Multivariate Anal. 46 154-174.
SERDOBOL'SKII, V. I. (1985). The resolvent and spectral functions of sample covariance matrices of increasing dimensions. Russian Math. Survey s 40 232-233.
Mathematical Reviews (MathSciNet): MR786104
SERDOBOL'SKII, V. I. (1995). Spectral properties of sample covariance matrices. Theory Probab. Appl. 40 777-786.
SERDOBOL'SKII, V. I. (1999). Theory of essentially multivariate statistical analysis. Russian Math. Survey s 54 351-379.
SILVERSTEIN, J. (1986). Eigenvalues and eigenvectors of large-dimensional sample covariance matrices. In Random Matrices and Their Applications (J. E. Cohen, H. Kesten and C. M. Newman, eds.) 153-159. Amer. Math. Soc., Providence, RI.
SILVERSTEIN, J. W. and COMBETTES, P. L. (1992). Signal detection via spectral theory of large dimensional random matrices. IEEE Trans. Signal Process. 40 2100-2105.
WACHTER, K. W. (1976). Probability plotting points for principal components. In Proceedings of the Ninth Interface Sy mposium on Computer Science and Statistics (D. Hoaglin and R. E. Welsch, eds.) 299-308. Prindle, Weber & Schmidt, Boston.
WACHTER, K. W. (1978). The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab. 6 1-18.
WILSON, W. J. and KSHIRSAGAR, A. M. (1980). An approach to multivariate analysis when the variance-covariance matrix is singular. Metron 38 81-92.
Mathematical Reviews (MathSciNet): MR652872
YIN, Y. Q. and KRISHNAIAH, P. R. (1983). A limit theorem for the eigenvalues of product of two random matrices. J. Multivariate Anal. 13 489-507.
Mathematical Reviews (MathSciNet): MR727035
ZHAO, L. C., KRISHNAIAH, P. R. and BAI, Z. D. (1986a). On detection of the number of signals in presence of white noise. J. Multivariate Anal. 20 1-25.
Mathematical Reviews (MathSciNet): MR862239
ZHAO, L. C., KRISHNAIAH, P. R. and BAI, Z. D. (1986b). On detection of the number of signals when the noise covariance matrix is arbitrary. J. Multivariate Anal. 20 26-49.
Mathematical Reviews (MathSciNet): MR862240
LOS ANGELES, CALIFORNIA 90095-1481 AND EQUITIES TRADING CREDIT SUISSE FIRST BOSTON ONE CABOT SQUARE LONDON E14 4QJ UNITED KINGDOM DEPARTMENT OF ECONOMICS AND BUSINESS UNIVERSITAT POMPEU FABRA
RAMON TRIAS FARGAS, 25-27 08005 BARCELONA SPAIN