The Annals of Applied Probability

Determinant of sample correlation matrix with application

Tiefeng Jiang

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Abstract

Let $\mathbf{x}_{1},\ldots ,\mathbf{x}_{n}$ be independent random vectors of a common $p$-dimensional normal distribution with population correlation matrix $\mathbf{R}_{n}$. The sample correlation matrix $\hat{\mathbf {R}}_{n}=(\hat{r}_{ij})_{p\times p}$ is generated from $\mathbf{x}_{1},\ldots ,\mathbf{x}_{n}$ such that $\hat{r}_{ij}$ is the Pearson correlation coefficient between the $i$th column and the $j$th column of the data matrix $(\mathbf{x}_{1},\ldots ,\mathbf{x}_{n})'$. The matrix $\hat{\mathbf {R}}_{n}$ is a popular object in multivariate analysis and it has many connections to other problems. We derive a central limit theorem (CLT) for the logarithm of the determinant of $\hat{\mathbf {R}}_{n}$ for a big class of $\mathbf{R}_{n}$. The expressions of mean and the variance in the CLT are not obvious, and they are not known before. In particular, the CLT holds if $p/n$ has a nonzero limit and the smallest eigenvalue of $\mathbf{R}_{n}$ is larger than $1/2$. Besides, a formula of the moments of $\vert \hat{\mathbf {R}}_{n}\vert $ and a new method of showing weak convergence are introduced. We apply the CLT to a high-dimensional statistical test.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 3 (2019), 1356-1397.

Dates
Received: October 2016
Revised: August 2017
First available in Project Euclid: 19 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1550566833

Digital Object Identifier
doi:10.1214/17-AAP1362

Mathematical Reviews number (MathSciNet)
MR3914547

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F05: Central limit and other weak theorems

Keywords
Central limit theorem sample correlation matrix smallest eigenvalue multivariate normal distribution moment generating function

Citation

Jiang, Tiefeng. Determinant of sample correlation matrix with application. Ann. Appl. Probab. 29 (2019), no. 3, 1356--1397. doi:10.1214/17-AAP1362. https://projecteuclid.org/euclid.aoap/1550566833


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References

  • Anderson, T. W. (1958). An Introduction to Multivariate Statistical Analysis, 2nd ed. Wiley, New York.
    Zentralblatt MATH: 0083.14601
  • Bai, Z. and Silverstein, J. W. (2010). Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer, New York.
    Zentralblatt MATH: 1301.60002
  • Bao, Z., Pan, G. and Zhou, W. (2012). Tracy–Widom law for the extreme eigenvalues of sample correlation matrices. Electron. J. Probab. 17 1–32.
    Zentralblatt MATH: 1254.15036
    Digital Object Identifier: doi:10.1214/EJP.v17-1962
  • Bartlett, M. S. (1954). A note on multiplying factors for various chi-squared approximations. J. R. Stat. Soc. Ser. B. Stat. Methodol. 16 296–298.
    Zentralblatt MATH: 0057.35404
    Digital Object Identifier: doi:10.1111/j.2517-6161.1954.tb00174.x
  • Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
    Zentralblatt MATH: 0649.60001
  • Brockwell, P. J. and Davis, R. A. (2002). Introduction to Time Series and Forecasting. Springer, New York.
    Zentralblatt MATH: 0994.62085
  • Cai, T., Fan, J. and Jiang, T. (2013). Distributions of angles in random packing on spheres. J. Mach. Learn. Res. 14 1837–1864.
    Zentralblatt MATH: 1318.60017
  • Cai, T., Liang, T. and Zhou, H. (2015). Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for high-dimensional Gaussian distributions. J. Multivariate Anal. 137 161–172.
    Zentralblatt MATH: 1329.62255
    Digital Object Identifier: doi:10.1016/j.jmva.2015.02.003
  • Chow, Y. S. and Teicher, H. (1988). Probability Theory: Independence, Interchangeability, Martingales, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR953964
    Zentralblatt MATH: 0652.60001
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
    Zentralblatt MATH: 0896.60013
  • Dong, Z., Jiang, T. and Li, D. (2012). Circular law and arc law for truncation of random unitary matrix. J. Math. Phys. 53 Article ID 013301.
    Zentralblatt MATH: 1273.15040
    Digital Object Identifier: doi:10.1063/1.3672885
  • Eaton, M. L. (1983). Multivariate Statistics: A Vector Space Approach. Wiley, New York.
    Zentralblatt MATH: 0587.62097
  • Hanson, D. L. and Wright, F. T. (1971). A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Stat. 42 1079–1083.
    Zentralblatt MATH: 0216.22203
    Digital Object Identifier: doi:10.1214/aoms/1177693335
    Project Euclid: euclid.aoms/1177693335
  • Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR832183
    Zentralblatt MATH: 0576.15001
  • Jiang, T. (2004a). The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Probab. 14 865–880.
    Mathematical Reviews (MathSciNet): MR2052906
    Zentralblatt MATH: 1047.60014
    Digital Object Identifier: doi:10.1214/105051604000000143
    Project Euclid: euclid.aoap/1082737115
  • Jiang, T. (2004b). The limiting distributions of eigenvalues of sample correlation matrices. Sankhyā 66 35–48.
    Mathematical Reviews (MathSciNet): MR2082906
    Zentralblatt MATH: 1193.62018
  • Jiang, T. and Qi, Y. (2015). Likelihood ratio tests for high-dimensional normal distributions. Scand. J. Stat. 42 988–1009.
    Mathematical Reviews (MathSciNet): MR3426306
    Zentralblatt MATH: 06527641
    Digital Object Identifier: doi:10.1111/sjos.12147
  • Jiang, T. and Yang, F. (2013). Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions. Ann. Statist. 41 2029–2074.
    Mathematical Reviews (MathSciNet): MR3127857
    Zentralblatt MATH: 1277.62149
    Digital Object Identifier: doi:10.1214/13-AOS1134
    Project Euclid: euclid.aos/1382547512
  • Li, D., Liu, W. and Rosalsky, A. (2010). Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix. Probab. Theory Related Fields 148 5–35.
    Zentralblatt MATH: 1210.62010
    Digital Object Identifier: doi:10.1007/s00440-009-0220-z
  • Li, D. and Rosalsky, A. (2006). Some strong limit theorems for the largest entries of sample correlation matrices. Ann. Appl. Probab. 16 423–447.
    Zentralblatt MATH: 1098.60034
    Digital Object Identifier: doi:10.1214/105051605000000773
    Project Euclid: euclid.aoap/1141654293
  • Morrison, D. F. (2004). Multivariate Statistical Methods, 4th ed. Duxbury Press, Pacific Grove, CA.
  • Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR652932
    Zentralblatt MATH: 0556.62028
  • Nguyen, H. H. and Vu, V. (2014). Random matrices: Law of the determinant. Ann. Probab. 42 146–167.
    Zentralblatt MATH: 1299.60005
    Digital Object Identifier: doi:10.1214/12-AOP791
    Project Euclid: euclid.aop/1389278522
  • Rudelson, M. and Vershynin, R. (2013). Hanson–Wright inequality and sub-Gaussian concentration. Electron. Commun. Probab. 18 Article ID 82.
    Zentralblatt MATH: 1329.60056
    Digital Object Identifier: doi:10.1214/ECP.v18-2865
  • Smale, S. (2000). Mathematical problems for the next century. In Mathematics: Frontiers and Perspectives (V. Arnold, M. Atiyah, P. Lax and B. Mazur, eds.) 271–294. Amer. Math. Soc., Providence, RI.
    Zentralblatt MATH: 1031.00005
  • Tao, T. and Vu, V. (2012). A central limit theorem for the determinant of a Wigner matrix. Adv. Math. 231 74–101.
    Zentralblatt MATH: 1248.60011
    Digital Object Identifier: doi:10.1016/j.aim.2012.05.006
  • Wilks, S. S. (1932). Certain generalizations in the analysis of variance. Biometrika 24 471–494.
    Zentralblatt MATH: 58.1172.02
    Digital Object Identifier: doi:10.1093/biomet/24.3-4.471
  • Yurinskiĭ, V. V. (1976). Exponential inequalities for sums of random vectors. J. Multivariate Anal. 6 473–499.
  • Zhou, W. (2007). Asymptotic distribution of the largest off-diagonal entry of correlation matrices. Trans. Amer. Math. Soc. 359 5345–5363.
    Zentralblatt MATH: 1130.60032
    Digital Object Identifier: doi:10.1090/S0002-9947-07-04192-X

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