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September, 1977 Some Variational Results and Their Applications in Multiple Inference
D. R. Jensen, L. S. Mayer
Ann. Statist. 5(5): 922-931 (September, 1977). DOI: 10.1214/aos/1176343948


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.


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D. R. Jensen. L. S. Mayer. "Some Variational Results and Their Applications in Multiple Inference." Ann. Statist. 5 (5) 922 - 931, September, 1977.


Published: September, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0368.62007
MathSciNet: MR448707
Digital Object Identifier: 10.1214/aos/1176343948

Primary: 62E10
Secondary: 62H05 , 62H15

Keywords: Monotone unitarily invariant functions , separation of singular values , Simultaneous confidence bounds , stochastic ordering , symmetric gauge functions

Rights: Copyright © 1977 Institute of Mathematical Statistics


Vol.5 • No. 5 • September, 1977
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