Hypothesis Testing for Independence Under Blocked Compound Symmetric Covariance Structure

Abstract

One type of covariance structure is known as blocked compound symmetry. Recently, Roy et al. (J Multivar Anal 144:81–90, 2016) showed that, assuming this covariance structure, unbiased estimators are optimal under normality and described hypothesis testing for independence as an open problem. In this paper, we derive the distributions of unbiased estimators and consider hypothesis testing for independence. Representative test statistics such as the likelihood ratio criterion, Wald statistic, Rao’s score statistic, and gradient statistic are derived, and we evaluate the accuracy of the test using these statistics through numerical simulations. The power of the Wald test is the largest when the dimension is high, and the power of the likelihood ratio test is the largest when the dimension is low.

Keywords

Hypothesis testing / Asymptotic distribution / Independence / Blocked compound symmetric covariance structure

References

[1] Anderson TW. An Introduction to Multivariate Statistical Analysis. 1984 2 New York: Wiley

[2] Arnold SF. Linear models with exchangeably distributed errors. J. Am. Stat. Assoc.. 1979, 74 194-199

[3] Coelho CA, Roy A. Testing the hypothesis of a block compound symmetric covariance matrix for elliptically contoured distributions. TEST. 2017, 26 308-330

[4] Efron B, Tibshirani RJ. An Introduction to the Bootstrap. 1994 New York: CRC Press

[5] Fujikoshi Y, Ulyanov VV, Shimizu R. Multivariate Statistics: High-Dimensional and Large-Sample Approximations. 2010 Hoboken: Wiley

[6] Gupta AK, Nagar DK. Matrix Variate Distributions. 1999 Boca Raton: Chapman and Hall

[7] Johnson RA, Wichern DW. Applied Multivariate Statistical Analysis. 2007 6 Englewood Cliffs: Pearson Prentice Hall

[8] Leiva R. Linear discrimination with equicorrelated training vectors. J. Multivar. Anal.. 2007, 98 384-409

[9] Pourahmadi M. High-Dimensional Covariance Estimation: With High-Dimensional Data. 2013 New York: Wiley

[10] Roy A, Leiva R. Estimating and testing a structured covariance matrix for three-level multivariate data. Commun. Stat. Theory Methods. 2011, 40 1945-1963

[11] Roy A, Leiva R, Žežula I, Klein D. Testing the equality of mean vectors for paired doubly multivariate observations in blocked compound symmetric covariance matrix setup. J. Multivar. Anal.. 2015, 137 50-60

[12] Roy A, Zmyślony R, Fonseca M, Leiva R. Optimal estimation for doubly multivariate data in blocked compound symmetric covariance structure. J. Multivar. Anal.. 2016, 144 81-90

[13] Szatrowski TH. Testing and estimation in the block compound symmetry problem. J. Educ. Stat.. 1982, 7 3-18

[14] Terrell GR. The gradient statistic. Comput. Sci. Stat.. 2002, 34 206-215

[15] Zezula I, Klein D, Roy A. Testing of multivariate repeated measures data with block exchangeable covariance structure. TEST. 2017