Some Variational Results and Their Applications in Multiple Inference

Abstract

Let $(\mathbf{M}, \mathbf{T}, \mathbf{S})$ be random matrices such that $\mathbf{M}$ and $\mathbf{S}$ are Hermitian positive definite almost everywhere. Let $\mathbf{M}_{(t)} = \lbrack m_{ij}; 1 \leqq i, j \leqq t\rbrack, \mathbf{S}_{(t)} = \lbrack s_{ij}; 1 \leqq i, j \leqq t\rbrack$ and $\mathbf{T}_{(r,s)} = \lbrack t_{ij}; 1 \leqq i \leqq r, 1 \leqq j \leqq s\rbrack$, and define $Q(r, s) = P\lbrack G((\mathbf{M}_{(r)})^{-\frac{1}{2}}\mathbf{T}_{(r,s)}(\mathbf{S}_{(s)}) ^{-\frac{1}{2}}) \leqq c\rbrack$ for some $G$ belonging to the class $\mathscr{G}$ of monotone unitarily invariant functions. The main result is that, for any $c$ and $G \in \mathscr{G}, Q(r, s)$ is a decreasing function of $r$ and $s$. Applications yield simultaneous confidence bounds for a variety of multivariate and multiparameter problems.