##
**Optimal hypothesis testing for high dimensional covariance matrices.**
*(English)*
Zbl 1281.62140

Summary: This paper considers testing a covariance matrix \(\Sigma\) in a high dimensional setting where the dimension \(p\) can be comparable or much larger than the sample size \(n\). The problem of testing the hypothesis \(H_{0}:\Sigma=\Sigma_{0}\) for a given covariance matrix \(\Sigma_{0}\) is studied from a minimax point of view. We first characterize the boundary that separates the testable regions from the non-testable regions by the Frobenius norm when the ratio between the dimension \(p\) over the sample size \(n\) is bounded. A test based on a \(U\)-statistic is introduced and is shown to be rate optimal in this asymptotic regime. Furthermore, it is shown that the power of this test uniformly dominates that of the corrected likelihood ratio test (CLRT) over the entire asymptotic regime under which the CLRT is applicable. The power of the \(U\)-statistic based test is also analyzed when \(p/n\) is unbounded.

### MSC:

62H15 | Hypothesis testing in multivariate analysis |

62C20 | Minimax procedures in statistical decision theory |

62F05 | Asymptotic properties of parametric tests |

### Keywords:

correlation matrix; high-dimensional data; likelihood ratio tests; minimax hypothesis testing; power; testing covariance structure
PDF
BibTeX
XML
Cite

\textit{T. T. Cai} and \textit{Z. Ma}, Bernoulli 19, No. 5B, 2359--2388 (2013; Zbl 1281.62140)

### References:

[1] | Anderson, T.W. (2003). An Introduction to Multivariate Statistical Analysis , 3rd ed. Wiley Series in Probability and Statistics . Hoboken, NJ: Wiley-Interscience [John Wiley & Sons]. · Zbl 1039.62044 |

[2] | Bai, Z., Jiang, D., Yao, J.F. and Zheng, S. (2009). Corrections to LRT on large-dimensional covariance matrix by RMT. Ann. Statist. 37 3822-3840. · Zbl 1360.62286 |

[3] | Birke, M. and Dette, H. (2005). A note on testing the covariance matrix for large dimension. Statist. Probab. Lett. 74 281-289. · Zbl 1070.62046 |

[4] | Cai, T.T. and Jiang, T. (2011). Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices. Ann. Statist. 39 1496-1525. · Zbl 1220.62066 |

[5] | Cai, T.T., Liu, W. and Xia, Y. (2011). Two-sample covariance matrix testing and support recovery. Technical report. · Zbl 06158341 |

[6] | Chen, S.X.and Li, J. (2012). Two sample tests for high dimensional covariance matrices. Ann. Statist. 40 908-940. · Zbl 1274.62383 |

[7] | Chen, S.X., Zhang, L.X. and Zhong, P.S. (2010). Tests for high-dimensional covariance matrices. J. Amer. Statist. Assoc. 105 810-819. · Zbl 1321.62086 |

[8] | Heyde, C.C. and Brown, B.M. (1970). On the departure from normality of a certain class of martingales. Ann. Math. Statist. 41 2161-2165. · Zbl 0225.60026 |

[9] | Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30. · Zbl 0127.10602 |

[10] | Jiang, D., Jiang, T. and Yang, F. (2012). Likelihood ratio tests for covariance matrices of high-dimensional normal distributions. J. Statist. Plann. Inference 142 2241-2256. · Zbl 1244.62082 |

[11] | Johnstone, I.M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295-327. · Zbl 1016.62078 |

[12] | Ledoit, O. and Wolf, M. (2002). Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. Ann. Statist. 30 1081-1102. · Zbl 1029.62049 |

[13] | Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. Wiley Series in Probability and Mathematical Statistics . New York: Wiley. · Zbl 0556.62028 |

[14] | Nagao, H. (1973). On some test criteria for covariance matrix. Ann. Statist. 1 700-709. · Zbl 0263.62034 |

[15] | Onatski, A., Moreira, M.J. and Hallin, M. (2011). Asymptotic power of sphericity tests for high-dimensional data. Available at . · Zbl 1293.62125 |

[16] | Roy, S.N. (1957). Some Aspects of Multivariate Analysis . New York: Wiley. · Zbl 0083.19305 |

[17] | Srivastava, M.S. (2005). Some tests concerning the covariance matrix in high dimensional data. J. Japan Statist. Soc. 35 251-272. |

[18] | Xiao, H. and Wu, W.B. (2011). Simultaneous inference on sample covariances. Available at . 1109.0524v1 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.