The Annals of Mathematical Statistics

Biorthogonal and Dual Configurations and the Reciprocal Normal Distribution

Robert H. Berk

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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.

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Ann. Math. Statist., Volume 40, Number 2 (1969), 393-398.

First available in Project Euclid: 27 April 2007

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Berk, Robert H. Biorthogonal and Dual Configurations and the Reciprocal Normal Distribution. Ann. Math. Statist. 40 (1969), no. 2, 393--398. doi:10.1214/aoms/1177697703.

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