Distributions of powers of the central beta matrix variates and applications
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Distributions of powers of the central beta matrix variates and applications Thu Pham-Gia1,4 · Duong Thanh Phong2,4 · Dinh Ngoc Thanh3,4 Accepted: 3 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We consider the central Beta matrix variates of both kinds, and establish the expressions of the densities of integral powers of these variates, for all their three types of distributions encountered in the statistical literature: entries, determinant, and latent roots distributions. Applications and computation of credible intervals are presented. Keywords Beta matrix variates · Credible interval · G-Function · Latent roots · Powers Mathematics Subject Classification 62H10
1 Introduction For a positive random variable X , its power X c , with c positive or negative, is often encountered in applications. “Weibullized” distribution is a particular application of this approach (see Bekker et al. 2009; Nadarajah and Kotz 2004; Pauw et al. 2010).
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Thu Pham-Gia [email protected] Duong Thanh Phong [email protected] Dinh Ngoc Thanh [email protected]
1
Département de Mathématiques et de Statistiques, Faculté des Sciences, Université de Moncton, Moncton, NB E1A3E9, Canada
2
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3
Faculty of Mathematics and Computer Science, University of Science, Vietnam National University Ho Chi Minh City, No. 227 Nguyen Van Cu Street, Ward 4, District 5, Ho Chi Minh City, Vietnam
4
Applied Multivariate Statistical Analysis Research Group, Ho Chi Minh City, Vietnam
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T. Pham-Gia et al.
Distributions with matrices as arguments have acquired increasing importance since Wishart (1928) established the distribution bearing his name in 1928, on the covariance matrix of a normal sample. The matrix Gamma, matrix Beta are derived from the Wishart, similarly to the univariate case, where we obtain the Beta from the univariate Normal through the Chi-square. But while there is only one distribution in the univariate case, there are at least three in R p , p ≥ 2, as explained in Sect. 3. To test hypotheses in multivariate analysis we have to use tools based on determinants, or latent roots, of certain matrices, among them the Betas, which are the subjects of this article. We consider first the univariate random variable U having central Beta distribution of the first kind, U ∼ Beta1I (a, b), where a, b > 0, with the probability density function (PDF): u a−1 (1 − u)b−1 , for 0 < u < 1, (1) f (u) = B (a, b) where B (a, b) = (a)(b) (a+b) , and the random variable V having central Beta distribution of the second kind, V ∼ Beta1I I (a, b), where a, b > 0, with PDF: g (v) =
v a−1 (1 + v)−a−b , for v > 0. B (a, b)
(2)
More generally, for some types of integral equations of Wilks type, Mathai (1984) has pointed out that the solutions involve some powers of Beta variables, as considered here. In the case of matrix variates, two types of Betas are also defined and play very important roles in Multivaria
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