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Summary The multivariate normal distribution can be used to describe the response variable of a system. A more comprehensive multivariate model is described in this paper: it has a distribution describing an error variable internal to a system, with a known multivariate distribution; and it has a positive affine transformation, the physical quantity, which generates a response vector from an error vector. This more comprehensive model is a structural model and it provides structural probability statements concerning the physical quantity. Error and structural distributions are derived for the multivariate model. The structural distribution for a quantity can be used to generate structural prediction distributions : various prediction distributions are obtained for the multivariate model. The results are specialized to cover the case of the multivariate normal structural model. The classical multivariate normal has been analysed by Bayesian methods (Geisser and Cornfield, 1963). The more comprehensive multivariate structural model does not need the use of subjective methods.

In the first part of the paper, we introduce the matrix-variate generalized hyperbolic distribution by mixing the matrix normal distribution with the matrix generalized inverse Gaussian density. The p-dimensional generalized hyperbolic distribution of [Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat., 5, 151-157], the matrix-T distribution and many well-known distributions are shown to be special cases of the new distribution. Some properties of the distribution are also studied. The second part of the paper deals with the application of the distribution in the Bayesian analysis of the normal multivariate linear model.

This paper derives the prediction distribution of future responses from the linear model with errors having an elliptical distribution with known covariance parameters. For unknown covariance parameters, the marginal likelihood function of the parameters has been obtained and the prediction distribution has been modified by replacing the covariance parameters by their estimates obtained from the marginal likelihood function. It is observed that the prediction distribution with elliptical error has a multivariate Student's t-distribution with appropriate degrees of freedom. The results for some special cases such as the Intra-class correlation model, AR(1), MA(1), and ARMA(1,1) models have been obtained from the general results. As an application, the ;-expectation tolerance region has been constructed. An example has been added. Academic Press