## The Annals of Statistics

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

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

#### 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