# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Goodman, N.R.: Statistical analysis based on a certain multivariate complex Gaussian distribution. An Introduction. Ann. Math. Statist.34, 1963, 152–176. · Zbl 0122.36903 · doi:10.1214/aoms/1177704250 [2] Halberstam, H., andR.E. Ingram: The Mathematical Papers of Sir William Hamilton. Vol. III. London 1967. [3] Kabe, D.G.: On some inequalities satisfied by beta and gamma functions. South African Statist J.12, 1978, 25–31. · Zbl 0383.62025 [4] Khatri, C.G.: Classical statistical analysis based on a certain multivariate complex Gaussian distribution., Ann. Math. Statist.36, 1965, 98–114. · Zbl 0135.19506 · doi:10.1214/aoms/1177700274 [5] Krishnaiah, P.R.: Some recent developments on complex multivariate distributions. J. Multi. Anal.6, 1976, 1–30. · Zbl 0358.62040 · doi:10.1016/0047-259X(76)90017-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.