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Average width optimality of simultaneous confidence bounds. (English) Zbl 0554.62029
Consider the usual linear regression model $$y=A\beta +e$$, where y and e are $$n\times 1$$, A is $$n\times k$$ of rank k, $$\beta$$ is $$k\times 1$$, and the elements of e are iid $$N(0,\sigma^ 2)$$. Simultaneous confidence intervals (CI’s) are wanted for all x’$$\beta$$, $$x\in X$$, in which X is some specified subset of $$R^ k$$ (e.g., $$R^ k$$, itself, or a bounded subset, or even a finite subset).
Only two-sided CI’s are considered with endpoints x’$${\hat \beta}\pm s\Phi (x)$$ or one-sided CI’s of the form (-$$\infty$$, x’$${\hat \beta}+s\Phi (x)]$$, in which $${\hat \beta}$$ is the least squares estimator of $$\beta$$, $$s^ 2$$ the usual estimator of $$\sigma^ 2$$, and $$\Phi$$ a function on X still to be chosen. With respect to a chosen measure $$\mu$$ on X the average width of the CI’s is defined as $$\int_{X}\Phi (x)\mu (dx)$$. The problem is to determine that $$\Phi$$, for given $$\mu$$, which minimizes the average width subject to the requirement that simultaneously (for all $$x\in X)$$ the CI’s cover their true values x’$$\beta$$ with probability at least 1-$$\alpha$$.
While that problem has not been solved, in the case that X is finite the paper gives a sufficient condition for $$\Phi$$ to be optimal relative to some $$\mu$$, and shows how to construct this $$\mu$$. Extension to infinite X by means of limits is discussed for simple linear regression with intercept. Also, the suboptimality is indicated of the Scheffé-type bounds when $$\mu$$ is Lebesgue measure on a sufficiently large finite interval.
Reviewer: R.A.Wijsman

##### MSC:
 62F25 Parametric tolerance and confidence regions 62J15 Paired and multiple comparisons; multiple testing 62J05 Linear regression; mixed models
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