Biorthogonal and Dual Configurations and the Reciprocal Normal Distribution

Abstract

In this note we discuss the notions of biorthogonal and dual configurations and their relevance in certain statistical applications. The first application is to the distribution of a random matrix related to a multi-variate-normal sample matrix. As with the latter, the distribution is preserved by (certain) linear transformations. One consequence of this is the familiar result that if $\mathbf{Q}$ is a non-singular Wishart matrix, then for any non-zero vector $\alpha, 1/\alpha'\mathbf{Q}^{-1}\alpha$ is a multiple of a chi-square variable. Application is also made to the Gauss-Markov theorem and to certain estimates of mixing proportions due to Robbins.