Classical statistical analysis based on a certain hypercomplex multivariate normal distribution

Abstract

Summary: Goodman [1963] generalized the real normal multivariate model to the complex case. Goodman [1963], and Khatri 119651 derived the sampling distribution theory underlying this model. The present paper generalizes the complex multivariate normal theory to the hypercomplex case. The hypercomplex case studied here includes Hamilton's quaternions, biquaternions, octonions, and bioctonions. It is shown that the complex case results straightforwardly generalize to the hypercomplex case.

On the real representation of quaternion random variables

“…We will make use of the representation theory throughout this paper, and although quaternions may be represented by real matrices in various ways, see for instance Teng and Fang [22], we will use the representation employed by Kabe (for instance [10] and [12]) and Rautenbach [19]. Specifically suppose that  =  1 +  2 +  3 +  4 ∈ Q may be represented by z 0 ∈  4 (R), as…”

“…In this section the approach of Kabe ([10], [11], [12]), Rautenbach [19] and Rautenbach and Roux ( [20], [21]) are followed in deriving the -variate quaternion normal distribution. Although the results in this section 3.1 are in general not new, it is shown how they relate to those given by Teng and Fang [22], and with particular emphasis on the quaternion and related real characteristic functions.…”

“…Let tr(·) be the trace on   (Q). Rautenbach and Roux [21] argued that the trace, as used by Kabe ([10], [11], [12]) in simplifying the exponent of   (), is not allowed, because of the noncommutativity property of quaternions, i.e. tr(AB) 6 =tr(BA) However, from (1), and the fact that for a quaternion random vector …”

“…Kabe ([10], [11], [12]) derived the hypercomplex Wishart distribution directly from the hypercomplex normal distribution using the  generalized Sverdrup's lemma. Teng and Fang [22] showed that the maximum likelihood estimatorΣ of Σ followed a quaternion Wishart distribution.…”

“…The first appearance of quaternion distributions in the statistical literature from the real representation approach is in papers of Kabe ([10], [11], [12]). Rautenbach [19] and Rautenbach and Roux ([20], [21]) also utilised the representation theory in their papers, and more recently there is the refreshing contribution by Teng and Fang [22].…”