Abstract
We derive expressions of statistics for testing covariance structures when the population is t-distributed. The likelihood ratio test, Rao’s score test and Wald’s score test are derived for basic covariance structures. Expressions of all three statistics are obtained under the general null-hypothesis H01 : Σ = Σ 0, using matrix derivative technique. Here p × p-matrix Σ is a dispersion/scale parameter. The special cases H02 : Σ = I p and H03 : Σ = γ 0I p where γ 0 > 0 is a known constant are also considered. Expressions of the statistics are obtained as approximations using first terms from Taylor expansions. The method can be carried over to other continuous multivariate elliptical distributions which have power function in the expression of the density function.
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References
Anderson, T.W.: An Introduction to Multivariate Statistical Analysis, 3rd edn. Wiley, New York (2003)
Bai, Z., Jiang, D., Yao, J., Zheng, S.: Corrections to LRT on large-dimensional covariance matrix by RMT. Ann. Statist. 37, 3822–3840 (2009)
Bilodeau, M., Brenner, D.: Theory of Multivariate Statistics. Springer, New York (1999)
Casella, G., Berger, R.L.: Statistical Inference. Duxbury Press, Belmont (1990)
Gombay, E.: Parametric sequential tests in the presence of nuisance parameters. Theory Stoch. Process. 8, 106–118 (2002)
Harville D.A.: Matrix Algebra from a Statistician’s Perspective. Springer, New York (1997)
Kollo, T.: A note on patterned matrices with applications in multivariate analysis. Acta Comment. Univ. Tartu. Math. 968, 17–27 (1994)
Kollo, T., von Rosen, D.: Advanced Multivariate Statistics with Matrices. Springer, Dordrecht (2005)
Kollo, T., von Rosen, D., Valge, M.: Hypotheses testing on covariance structures: comparison of likelihood ratio test, Rao’s score test and Wald’s score test. In: Bozeman, J.R., Oliveira, T., Skiadas, C.H. (eds.) Stochastic and Data Analysis Methods and Applications in Statistics and Demography (ISAST), pp. 423–435 (2016)
Kotz, S., Nadarajah, S.: Multivariate t Distributions and their Applications. Cambridge University, Cambridge (2004)
Magnus, J., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics, Revised edn. Wiley, New York (1999)
Muirhead, R.J.: Aspects of Multivariate Statistical Theory. 2nd edn. Wiley, Hoboken (2005)
Rao, C.R.: Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation. Math. Proc. Cambridge Philos. Soc. 44, 50–57 (1948)
Rao, C.R.: Linear Statistical Inference and Its Applications. 2nd edn. Wiley, New York (1973)
Srivastava, M.S., von Rosen, T., von Rosen, D.: Estimation and testing in general multivariate linear models with Kronecker product covariance structure. Sankhyā Ser. A 71, 137–163 (2009)
Wald, A.: Tests of statistical hypotheses concerning several parameters, when the number of observations is large. Trans. Amer. Math. Soc. 54, 426–482 (1943)
Yuan, K.-H., Bentler, P.M.: On normal theory and associated test statistics in covariance structure analysis under two classes of nonnormal distributions. Statist. Sinica 9, 831–853 (1999)
Acknowledgements
The authors are thankful to the Referee for valuable comments which helped to avoid some misprints and improved presentation of the paper. Tõnu Kollo is grateful to Professor Dietrich von Rosen for discussions on different topics of multivariate statistics and inspiring ideas during cooperation over 30 years. T. Kollo and M. Valge are grateful for financial support from institutional target financed project IUT34-5. M. Valge is also thankful for financial support from Estonian Mathematics and Statistics Doctoral School.
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Kollo, T., Valge, M. (2020). Covariance Structure Tests for t-distribution. In: Holgersson, T., Singull, M. (eds) Recent Developments in Multivariate and Random Matrix Analysis. Springer, Cham. https://doi.org/10.1007/978-3-030-56773-6_12
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DOI: https://doi.org/10.1007/978-3-030-56773-6_12
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