## The Annals of Mathematical Statistics

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

### 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.

#### 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