The Conditional Wishart: Normal and Nonnormal

Abstract

The response variable of a general multivariate model can be constructed as a positive affine transformation of a vector error variable. In the case of an error variable that is rotationally symmetric, the multivariate model has parameters that can be expressed as the mean vector and the variance matrix. In the case of an error variables that is standard normal, it becomes the ordinary multivariate normal. For the general multivariate model with rotationally-symmetric error the sample inner-product matrix is a conventional statistic for inference. The distribution of this statistics is derived: the distribution is a conditional distribution given observable characteristics of the error variable. The response variable of a more specialized multivariable model can be constructed as a positive linear transformation of a vector error variable. In the case of an error variable that is rotationally symmetric, the model has parameters that can be expressed in terms of the variance matrix, the mean vector being zero. In the case of an error variable that is standard normal; it becomes the ordinary central multivariate normal. For the multivariate model with rotationally-symmetric error, the distribution of the Wishart matrix is derived; the distribution is a conditional distribution given observable characteristics of the error variable. And for the multivariate model with standard normal error, the noncentral distribution of the Wishart matrix is also derived, again as the appropriate conditional distribution.