Some Variational Results and Their Applications in Multiple Inference
| Autor: | L. S. Mayer, D. R. Jensen |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 1977 |
| Předmět: |
Statistics and Probability
Mathematical optimization 62E10 symmetric gauge functions Inference Stochastic ordering stochastic ordering separation of singular values Singular solution 62H05 Applied mathematics 62H15 Statistics Probability and Uncertainty Monotone unitarily invariant functions simultaneous confidence bounds Mathematics |
| Zdroj: | Ann. Statist. 5, no. 5 (1977), 922-931 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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