The Annals of Statistics

Constructing All Smallest Simultaneous Confidence Sets in a Given Class with Applications to MANOVA

Robert A. Wijsman

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Abstract

A method is presented for the construction of all families of smallest simultaneous confidence sets (SCS) in a given class, for a family $\{\psi_i(\gamma)\}$ of parametric functions of the parameter of interest $\gamma = \gamma(\theta)$. The method is applied to the MANOVA problem (in its canonical form) of inference about $M = EX$, where $X$ is $q \times p$ and has rows that are independently multivariate normal with common covariance matrix $\Sigma$. Let $S$ be the usual estimate of $\Sigma$ and put $W = (M - X)S^{-\frac{1}{2}}$. It is shown that smallest equivariant SCS for all $a'M, a \in R^q$, are necessarily those that are exact with respect to the confidence set for $M$ determined by $\lambda_1(WW') \leqslant \operatorname{const} (\lambda_1 = \text{maximum characteristic root})$, i.e., derived from the acceptance region of Roy's maximum root test (this is strictly true for $p < q$, and true for $p \geqslant q$ under a weak additional restriction). It is also shown that smallest equivariant SCS for all tr $NM$, with rank $(N) \leqslant r$, are necessarily those that are exact with respect to $\|W\|_{\varphi_r} \leqslant 1$, where $\varphi_r$ is a symmetric gauge function that, on the ordered positive cone, depends only on the first $r$ arguments. Taking $r = 1$, the simultaneous confidence intervals for all $a'Mb$ of Roy and Bose emerge, and $r = \min(p, q)$ results in the simultaneous confidence intervals for all tr $NM$ of Mudholkar.

Article information

Source
Ann. Statist., Volume 7, Number 5 (1979), 1003-1018.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176344784

Digital Object Identifier
doi:10.1214/aos/1176344784

Mathematical Reviews number (MathSciNet)
MR536503

Zentralblatt MATH identifier
0416.62030

JSTOR
links.jstor.org

Subjects
Primary: 62F25: Tolerance and confidence regions
Secondary: 62J10: Analysis of variance and covariance 15A45: Miscellaneous inequalities involving matrices

Keywords
Simultaneous confidence sets simultaneous confidence intervals smallest exact equivariant MANOVA symmetric gauge functions extremum lemmas involving trace

Citation

Wijsman, Robert A. Constructing All Smallest Simultaneous Confidence Sets in a Given Class with Applications to MANOVA. Ann. Statist. 7 (1979), no. 5, 1003--1018. doi:10.1214/aos/1176344784. https://projecteuclid.org/euclid.aos/1176344784


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