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.