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Classical statistical analysis based on a certain hypercomplex multivariate normal distribution. (English) Zbl 0537.62039
N. R. Goodman [Ann. Math. Statistics 34, 152-177 (1963; Zbl 0122.369)] derived the sampling distribution theory underlying the complex normal multivariate model. In the present paper the author further generalizes this theory to the hypercomplex case. He defines the hypercomplex (HC) normal distribution as follows. Let \(X_ 1,X_ 2,...,X_{4t}\) be 4t real random matrices, each \(p\times N\) dimensional. Let t be a discrete parameter which can take on the values 1/4, 1/2, 1, 2. Then \(X^ T=[X^ T\!_ 1,...,X^ T\!_{4t}]\) is said to have a HC normal distribution if its density function is \[ (4\pi t)^{-2pNt}| \Sigma_ 0|^{-frac{1}{2}N}\exp [-tr(X-\mu)(X-\mu)^ T/4t] \] and \(\Sigma_ 0\) and \(\mu\) have a certain structure. When \(t=2\) we have the octonion case, when \(t=1\) we have the quaternion case, when \(t=frac{1}{2}\) we have the complex case, and when \(t=1/4\) we have the real case. The matrix \(\Sigma_ 0\) is an 8\(\times 8\) partitioned matrix (for \(t=2).\)
For \(t=1\), \(\Sigma_ 0\) is the 4\(\times 4\) partitioned matrix in the upper right-hand corner of \(\Sigma_ 0\) for \(t=2\). For \(t=frac{1}{2}\), \(\Sigma_ 0\) is the 2\(\times 2\) partitioned matrix in the upper right- hand corner of \(\Sigma_ 0\) for \(t=2\). For \(t=1/4\), \(\Sigma_ 0\) is the 1\(\times 1\) partitioned matrix in the upper right-hand corner of \(\Sigma_ 0\) for \(t=2\). Some interpretation is necessary. For example, if \(p=1\), then in the real case \(\Sigma_ 0\) is not the variance of the components of X.
Reviewer: K.S.Miller

MSC:
62H10 Multivariate distribution of statistics
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References:
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