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

Robert A. Wijsman
The Annals of Statistics
Vol. 7, No. 5 (Sep., 1979), pp. 1003-1018
Stable URL: http://www.jstor.org/stable/2958669
Page Count: 16
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Constructing All Smallest Simultaneous Confidence Sets in a Given Class with Applications to MANOVA
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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 {ψi(γ)} of parametric functions of the parameter of interest γ = γ(θ). The method is applied to the MANOVA problem (in its canonical form) of inference about M = EX, where X is q × p and has rows that are independently multivariate normal with common covariance matrix Σ. Let S be the usual estimate of Σ and put W = (M - X)S-1/2. It is shown that smallest equivariant SCS for all a'M, a ∈ Rq, 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 ⩾ q under a weak additional restriction). It is also shown that smallest equivariant SCS for all tr NM, with rank (N) ⩽ r, are necessarily those that are exact with respect to |W|φr ⩽ 1, where φr 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.

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