Random Quotients and the Behrens-Fisher Problem

Abstract

Let $\mathscr{P}_n$ be the space of $n \times n$ positive definite symmetric matrices. If $S_1$ and $S_2$ are random matrices in $\mathscr{P}_n, S_1$ is a better $\alpha$ denominator than $S_2$ (written $S_1 \prec_{(\alpha)} S_2$) $\operatorname{iff} U(x'S_1^{-1}x)^{\alpha/2} \ll_{st} U(x'S_2^{-1}x)^{\alpha/2}$ for all $x \in R^n$ where $U$ is uniform on [0, 1], independent of $S_1$ and $S_2, \alpha > 0$, and "$\ll_{st}$" means stochastically smaller than. A principal result is this. THEOREM. Let $S_1, \cdots, S_m$ be exchangeable random matrices in $\mathscr{P}_n$. If $0 < \alpha \leqq 2$, then $\sum^m_{i=1} \eta_i S_i \prec_{(\alpha)} \sum^m_{i=1} \psi_iS_i$ provided $(\psi_1, \cdots, \psi_m)$ majorizes $(\eta_1, \cdots, \eta_m)$. This has applications in establishing probability inequalities for certain common test statistics. The results in this paper extend those of Lawton. (Some inequalities for central and non-central distributions. Ann. Math. Statist. (1965) 36 1521-1525; Concentration of random quotients. Ann. Math. Statist. (1968) 39 466-480.)