The Joint Distribution of Traces of Wishart Matrices and Some Applications

Abstract

Let $\mathbf{W}_{jj}$ and $\mathbf{\Sigma}_{jj}, 1 \leqq j \leqq q$, respectively denote the diagonal blocks of a partitioned Wishart matrix $\mathbf{W}$ and its matrix $\mathbf{\Sigma}$ of parameters. A Laguerrian expansion is given for the joint distribution of $v_j = \mathrm{tr} \mathbf{W}_{jj}\mathbf{\Sigma}^{-1}_{jj}, 1 \leqq j \leqq q$, which is a generalization of known multivariate chi-square distributions. Approximations to the joint distribution function are discussed, and probability inequalities are given for this and a related multivariate $F$-distribution. Applications are made to some simultaneous multivariate test procedures.