The Role of the Real Representation – In Quaternion Distribution Theory
M.T. Loots1, A. Bekker1, M. Arashi2, J.J.J. Roux1
1Faculty of Natural and Agricultural Sciences, Department of Statistics, University of Pretoria, Pretoria, South Africa; 2Faculty of Mathematics, Shahrood University of Technology, Shahrood, Islamic Republic of Iran

Although quaternions had been invented during the first half of the 19th century, they only made their first appearance in the statistical literature in an article by Andersson (1975). Andersson employed an indirect approach in his development of the quaternion normal distribution by imposing conditions on its expected value and covariance. However, both Rautenbach (1983), and Teng and Fang (1997), independently remarked that certain aspects underlying the quaternion distribution theory are lost, or not stated explicitly, when working with these invariant normal models. Kabe (1976), (1978), (1984), generalised the work done by Goodman (1963), and Khatri (1965), from the complex to the hypercomplex space. Kabe's approach utilised the representation theory, and was further studied by Rautenbach (1983), Rautenbach and Roux (1983), (1985), and more recently by Teng and Fang (1997). First, we devote our attention to a review on the derivation of the p-variate quaternion normal distribution using the representation theory, and thereafter, the matrix-variate quaternion normal distribution is derived by generalising this approach. We note that the quaternion chi-squared distribution reduces to an associated real-valued form, and although the quaternion Wishart distribution is not a new addition to the family of matrix-variate quaternion distributions, it will be shown how the quaternion Wishart distribution relates to its real counterpart. Applications of this real representation approach in distribution theory are illustrated.

References:

[1] S. Andersson. Invariant normal models. The Annals of Statistics, 3(1):132–154, 1975.

[2] N.R. Goodman. Statistical analysis based on a certain multivariate complex gaussian distribution (an introduction). Annals of Mathematical Statistics, 34:152–177, 1963.

[3] D.G. Kabe. Q generalized Sverdrup's lemma. The Canadian Journal of Statistics/La Revue Canadienne de Statistique, 4(2):221–226, 1976.

[4] D.G. Kabe. On some inequalities satisfied by beta and gamma functions. Suid-Afrikaanse Statistiese Tydskrif, 12:25–31, 1978.

[5] D.G. Kabe. Classical statistical analysis based on a certain hypercomplex multivariate normal distribution. Metrika, 31(1):63–76, 1984.

[6] C.G. Khatri. Classical statistical analysis based on a certain multivariate complex gaussian distribution. The Annals of Mathematical Statistics, 36(1):98–114, 1965.

[7] H.M. Rautenbach. Kwaternione – Ontwikkeling met Spesiale Verwysing na die Kwaternioonnormaalverdeling. Doctoral thesis, University of South Africa, University of South Africa, Department of Statistics, South Africa, June 1983. Unpublished doctoral thesis.

[8] H.M Rautenbach and J.J.J. Roux. Mathematical concepts for quaternions (wiskundige begrippe vir kwaternione). Technical Report 83/7, University of South Africa, Pretoria, 1983.

[9] H.M Rautenbach and J.J.J. Roux. Statistical analysis based on quaternion normal random variables. Technical Report 85/9, University of South Africa, Pretoria, 1985.

[10] C.Y. Teng and K.T. Fang. Statistical analysis based on normal distribution of quaternion. 1997 International Symposium on Contemporary Multivariate Analysis and Its Applications, May 1997.

Keywords: quaternion matrix-variate beta type I distribution; quaternion normal distribution; quaternion Wishart distribution; real representation theory

Biography: Theodor Loots is an active member of the “Beyond known distributions” research group at the Department of Statistics in the University of Pretoria, South Africa, under the leadership of Prof. A. Bekker. His research interests revolve around distribution theory, not limited to, but including quaternion distribution theory, descriptive measures and alternative representations of “known” results.