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Keywords:
high-dimensional inference; covariance testing; $U$-statistics; non-normality
high-dimensional inference; covariance testing; $U$-statistics; non-normality
Summary:
A test statistic for homogeneity of two or more covariance matrices is presented when the distributions may be non-normal and the dimension may exceed the sample size. Using the Frobenius norm of the difference of null and alternative hypotheses, the statistic is constructed as a linear combination of consistent, location-invariant, estimators of trace functions that constitute the norm. These estimators are defined as $U$-statistics and the corresponding theory is exploited to derive the normal limit of the statistic under a few mild assumptions as both sample size and dimension grow large. Simulations are used to assess the accuracy of the statistic.
References:
[1] Ahmad, M. R.: Location-invariant multi-sample $U$-tests for covariance matrices with large dimension. Scand. J. Stat. 44 (2017b), 500-523. DOI 10.1111/sjos.12262 | MR 3658524
[2] Ahmad, M. R.: Testing homogeneity of several covariance matrices and multi-sample sphericity for high-dimensional data under non-normality. Comm. Stat. Theory Methods 46 (2017a), 3738-3753. DOI 10.1080/03610926.2015.1073310 | MR 3590835
[3] Ahmad, M. R.: On testing sphericity and identity of a covariance matrix with large dimensions. Math. Meth. Stat. 25 (2016), 121-132. DOI 10.3103/s1066530716020034 | MR 3519645
[4] Anderson, T.W.: An Introduction to Multivariate Statistical Analysis. Third edition. Wiley, NY 2003. MR 1990662
[5] Aoshima, M., Yata, K.: Two-stage procedures for high-dimensional data. Seq. An. 30 (2011), 356-399. DOI 10.1080/07474946.2011.619088 | MR 2855952
[6] Cai, T., Liu, W., Xia, Y.: Two-sample covariance matrix testing and support recovery in high-dimensional and sparse settings. J. Amer. Statist. Assoc. 108 (2013), 265-277. DOI 10.1080/01621459.2012.758041 | MR 3174618
[7] Fujikoshi, Y., Ulyanov, V. V., Shimizu, R.: Multivariate statistics: High-dimensional and large-sample approximations. Wiley, NY 2010. MR 2640807
[8] Hájek, J., Šidák, Z., Sen, P. K.: Theory of Rank Tests. Academic Press, SD 1999. MR 1680991 | Zbl 0944.62045
[9] Kim, T. Y., Luo, Z-M., Kim, C.: The central limit theorem for degenerate variable $U$-statistics under dependence. J. Nonparam. Stat. 23 (2011), 683-699. DOI 10.1080/10485252.2011.556193 | MR 2836284
[10] Koroljuk, V. S., Borovskich, Y. V.: Theory of $U$-statistics. Kluwer Academic Press, Dordrecht 1994. DOI 10.1007/978-94-017-3515-5 | MR 1472486
[11] Lee, A. J.: $U$-statistics: Theory and Practice. CRC Press, Boca Raton 1990.
[12] Lehmann, E. L.: Elements of Large-sample Theory. Springer, NY 1999. DOI 10.1007/b98855 | MR 1663158
[13] Li, J., Chen, S. X.: Two sample tests for high-dimensional covariance matrices. Ann. Stat. 40 (2012), 908-940. DOI 10.1214/12-aos993 | MR 2985938
[14] Liu, B., Xu, L., Zheng, S., Tian, G-L.: A new test for the proportionality of two large-dimensional covariance matrices. J. Multiv. An. 131 (2014), 293-308. DOI 10.1016/j.jmva.2014.06.008 | MR 3252651
[15] Mikosch, T.: Weak invariance principles for weighted $U$-statistics. J. Theoret. Prob. 7 (1991), 147-173. DOI 10.1007/bf02213365 | MR 1256396
[16] Mikosch, T.: A weak invariance principle for weighted $U$-statistics with varying kernels. J. Multiv. An. 47 (1993), 82-102. DOI 10.1006/jmva.1993.1072 | MR 1239107
[17] Muirhead, R. J.: Aspects of Multivariate Statistical Theory. Wiley, NY 2005 DOI 10.1002/9780470316559 | MR 0652932
[18] Pinheiro, A., Sen, K., Pinheiro, H. P.: Decomposibility of high-dimensional diversity measures: Quasi-$U$-statistics, martigales, and nonstandard asymptotics. J. Multiv. An. 100 (2009), 1645-1656. DOI 10.1016/j.jmva.2009.01.007 | MR 2535376
[19] Qiu, Y., Chen, S. X.: Test for bandedness of high-dimensional covariance matrices and bandwidth estimation. Ann. Stat. 40 (2012) 1285-1314. DOI 10.1214/12-aos1002 | MR 3015026
[20] Schott, J. R.: A test for the equality of covariance matrices when the dimension is large relative to the sample size. Computat. Statist. Data Analysis 51 (2007), 6535-6542. DOI 10.1016/j.csda.2007.03.004 | MR 2408613
[21] Seber, G. A. F.: Multivariate Observations. Wiley, NY 2004. DOI 10.1002/9780470316641 | MR 0746474
[22] Sen, P. K.: Robust statistical inference for high-dimensional data models with applications in genomics. Aust. J. Stat. 35 (2006), 197-214.
[23] Serfling, R. J.: Approximation Theorems of Mathematical Statistics. Wiley, Weinheim 1980. DOI 10.1002/9780470316481 | MR 0595165
[24] Srivastava, M. S., Yanagihara, H.: Testing the equality of several covariance matrices with fewer observations than the dimension. J. Multiv. An. 101, 1319-1329. DOI 10.1016/j.jmva.2009.12.010 | MR 2609494
[25] Vaart, A. W. van der: Asymptotic Statistics. Cambridge University Press, 1998. DOI 10.1017/cbo9780511802256 | MR 1652247
[26] Zhong, P-S., Chen, S. X.: Tests for high-dimensional regression coefficients with factorial designs. J. Amer. Statist. Assoc. 106 (2011), 260-274. DOI 10.1198/jasa.2011.tm10284 | MR 2816719
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