zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. (English) Zbl 1029.62049
Summary: 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 properties and limiting distributions 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.

62H15Multivariate hypothesis testing
62E20Asymptotic distribution theory in statistics
Full Text: DOI Euclid
[1] ALALOUF, I. S. (1978). An explicit treatment of the general linear model with singular covariance matrix. Sankhy?a Ser. B 40 65-73. · Zbl 0427.62049
[2] ANDERSON, T. W. (1984). An Introduction to Multivariate Statistical Analy sis, 2nd ed. Wiley, New York.
[3] ARHAROV, L. V. (1971). Limit theorems for the characteristic roots of a sample covariance matrix. Soviet Math. Dokl. 12 1206-1209. · Zbl 0239.62018
[4] BAI, Z. D. (1993). Convergence rate of expected spectral distributions of large random matrices. II. Sample covariance matrices. Ann. Probab. 21 649-672. · Zbl 0779.60024 · doi:10.1214/aop/1176989261
[5] 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. · Zbl 0677.94001 · doi:10.1109/18.32132
[6] BAI, Z. D. and SARANADASA, H. (1996). Effect of high dimension: By an example of a two sample problem. Statist. Sinica 6 311-329. · Zbl 0848.62030
[7] COHEN, A. and STRAWDERMAN, W. E. (1971). Unbiasedness of tests for homogeneity of variances. Ann. Math. Statist. 42 355-360. · Zbl 0218.62082 · doi:10.1214/aoms/1177693520
[8] DEMPSTER, A. P. (1958). A high dimensional two sample significance test. Ann. Math. Statist. 29 995-1010. · Zbl 0226.62014 · doi:10.1214/aoms/1177706437
[9] DEMPSTER, A. P. (1960). A significance test for the separation of two highly multivariate small samples. Biometrics 16 41-50. JSTOR: · Zbl 0218.62065 · doi:10.2307/2527954 · http://links.jstor.org/sici?sici=0006-341X%28196003%2916%3A1%3C41%3AASTFTS%3E2.0.CO%3B2-1&origin=euclid
[10] GIRKO, V. L. (1979). The central limit theorem for random determinants. Theory Probab. Appl. 24 729-740. · Zbl 0416.60028
[11] GIRKO, V. L. (1988). Spectral Theory of Random Matrices. Nauka, Moscow (in Russian). · Zbl 0656.15012
[12] GLESER, L. J. (1966). A note on the sphericity test. Ann. Math. Statist. 37 464-467. · Zbl 0138.13901 · doi:10.1214/aoms/1177699529
[13] JOHN, S. (1971). Some optimal multivariate tests. Biometrika 58 123-127. JSTOR: · Zbl 0218.62055 · http://links.jstor.org/sici?sici=0006-3444%28197104%2958%3A1%3C123%3ASOMT%3E2.0.CO%3B2-M&origin=euclid
[14] JOHN, S. (1972). The distribution of a statistic used for testing sphericity of normal distributions. Biometrika 59 169-173. JSTOR: · Zbl 0231.62072 · doi:10.1093/biomet/59.1.169 · http://links.jstor.org/sici?sici=0006-3444%28197204%2959%3A1%3C169%3ATDOASU%3E2.0.CO%3B2-6&origin=euclid
[15] JONSSON, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 1-38. · Zbl 0491.62021 · doi:10.1016/0047-259X(82)90080-X
[16] LÄUTER, J. (1996). Exact t and F tests for analyzing studies with multiple endpoints. Biometrics 52 964-970. · Zbl 0867.62049 · doi:10.2307/2533057
[17] 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. · Zbl 0162.22501 · doi:10.1070/SM1967v001n04ABEH001994
[18] MARSHALL, A. W. and OLKIN, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York. · Zbl 0437.26007
[19] MUIRHEAD, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York. · Zbl 0556.62028
[20] NAGAO, H. (1973). On some test criteria for covariance matrix. Ann. Statist. 1 700-709. · Zbl 0263.62034 · doi:10.1214/aos/1176342464
[21] NARAy ANASWAMY, C. R. and RAGHAVARAO, D. (1991). Principal component analysis of large dispersion matrices. Appl. Statist. 40 309-316. JSTOR: · Zbl 0825.62523 · doi:10.2307/2347595 · http://links.jstor.org/sici?sici=0035-9254%281991%2940%3A2%3C309%3APCAOLD%3E2.0.CO%3B2-R&origin=euclid
[22] 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. · Zbl 0778.62055 · doi:10.1006/jmva.1993.1054
[23] SERDOBOL’SKII, V. I. (1985). The resolvent and spectral functions of sample covariance matrices of increasing dimensions. Russian Math. Survey s 40 232-233. · Zbl 0582.60045 · doi:10.1070/RM1985v040n02ABEH003575
[24] SERDOBOL’SKII, V. I. (1995). Spectral properties of sample covariance matrices. Theory Probab. Appl. 40 777-786. · Zbl 0898.62081
[25] SERDOBOL’SKII, V. I. (1999). Theory of essentially multivariate statistical analysis. Russian Math. Survey s 54 351-379. · Zbl 1061.62531 · doi:10.1070/rm1999v054n02ABEH000133
[26] 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. · Zbl 0586.60033
[27] 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.
[28] 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.
[29] WACHTER, K. W. (1978). The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab. 6 1-18. · Zbl 0374.60039 · doi:10.1214/aop/1176995607
[30] WILSON, W. J. and KSHIRSAGAR, A. M. (1980). An approach to multivariate analysis when the variance-covariance matrix is singular. Metron 38 81-92. · Zbl 0473.62046
[31] 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. · Zbl 0553.62018 · doi:10.1016/0047-259X(83)90035-0
[32] 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. · Zbl 0617.62055 · doi:10.1016/0047-259X(86)90017-5
[33] 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. · Zbl 0617.62056 · doi:10.1016/0047-259X(86)90018-7