On a Paradox Concerning Inference About a Convariance Matrix

Abstract

Suppose a $p \times p$ dispersion matrix $\mathbf{T}$ is considered to have the Wishart distribution $W (\Sigma, n)$, c. f. Anderson (1958) p. 158, where $\Sigma$ is an arbitrary full rank covariance matrix and $n \geqq p$. Suppose $\mathbf{T}$ is observable but $\Sigma$ is unknown, and suppose a posterior distribution is to be assigned to $\Sigma$ given $\mathbf{T}$, where the term posterior is meant in a wide sense to allow the use of a Bayesian or fiducial or any other form of reasoning in arriving at the posterior distribution. A lemma is proved in Section 3 giving a property of all such posterior distributions which possess a natural linear invariance property. The paradoxical nature of this property is discussed in Section 4.