Abstract
While the likelihood ratio test for the equality of mean vectors, when the covariance matrices are assumed to be only positive-definite, is a common test in multivariate analysis, similar likelihood ratio tests are not available in the literature when the covariance matrices are assumed to have some common given structure. In this compact paper the author deals with the problem of developing likelihood ratio tests for the equality of mean vectors when the covariance matrices are assumed to have a circular or circulant structure. The likelihood ratio statistic is obtained and its exact distribution is expressed in terms of products of independent Beta random variables. Then, it is shown how for some particular cases it is possible to obtain very manageable finite form expressions for the probability density and cumulative distribution functions of this distribution, while for the other cases, given the intractability of the expressions for these functions, very sharp near-exact distributions are developed. Numerical studies show the extreme closeness of these near-exact distributions to the exact distributions.
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References
Anderson, T.W.: The Statistical Analysis of Time Series. Wiley, New York (1972)
Anderson, T.W.: An Introduction to Multivariate Statistical Analysis, 3rd edn. Wiley, Hoboken (2003)
Arnold, B.C., Coelho, C.A., Marques, F.J.: The distribution of the product of powers of independent uniform random variables – a simple but useful tool to address and better understand the structure of some distributions. J. Multivar. Anal. 113, 19–36 (2013)
Coelho, C.A.: The generalized integer Gamma distribution – a basis for distributions in multivariate statistics. J. Multivar. Anal. 64, 86–102 (1998)
Coelho, C.A.: The generalized near-integer Gamma distribution: a basis for ‘near-exact’ approximations to the distribution of statistics which are the product of an odd number of independent Beta random variables. J. Multivar. Anal. 89, 191–218 (2004)
Coelho, C.A., Arnold, B.C., Marques, F.J.: Near-exact distributions for certain likelihood ratio test statistics. J. Stat. Theory Pract. 4, 711–725 (2010)
John, J.A.: Cyclic Designs. Chapman and Hall, London (1987)
Khattree, R.: Multivariate statistical inference involving circulant matrices: a review. In: Gupta, A.K., Girko, V.L. (eds.) Multidimensional Statistical Analysis and Theory of Random Matrices, Proceedings of the Sixth Lukacs Symposium, pp. 101–110. VSP, The Nederlands (1996)
Kshirsagar, A.M.: Multivariate Analysis. Marcel Dekker, New York (1972)
Marques, F.J., Coelho, C.A.: Obtaining the exact and near-exact distributions of the likelihood ratio statistic to test circular symmetry through the use of characteristic functions. Comput. Stat. 28, 2091–2115 (2013)
Marques, F.J., Coelho, C.A., Arnold, B.C.: A general near-exact distribution theory for the most common likelihood ratio test statistics used in multivariate analysis. TEST 20, 180–203 (2011)
Muirhead, R.J.: Aspects of Multivariate Statistical Theory, 2nd edn. Wiley, Hoboken (2005)
Olkin, I., Press, S.J.: Testing and estimation for a circular stationary model. Ann. Math. Stat. 40, 1358–1373 (1969)
Pollock, D.S.G.: Circulant matrices and time-series analysis. Int. J. Math. Educ. Sci. Technol. 33, 213–230 (2002)
Tricomi, F.G., Erdélyi, A.: The asymptotic expansion of a ratio of Gamma functions. Pac. J. Math. 1, 133–142 (1951)
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Coelho, C.A. (2018). Likelihood Ratio Tests for Equality of Mean Vectors with Circular Covariance Matrices. In: Oliveira, T., Kitsos, C., Oliveira, A., Grilo, L. (eds) Recent Studies on Risk Analysis and Statistical Modeling. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-76605-8_18
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DOI: https://doi.org/10.1007/978-3-319-76605-8_18
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