The Annals of Mathematical Statistics

Biorthogonal and Dual Configurations and the Reciprocal Normal Distribution

Robert H. Berk

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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.

Article information

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

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177697703

Digital Object Identifier
doi:10.1214/aoms/1177697703

Mathematical Reviews number (MathSciNet)
MR238433

Zentralblatt MATH identifier
0174.22307

JSTOR
links.jstor.org

Citation

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. https://projecteuclid.org/euclid.aoms/1177697703


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