Average width optimality of simultaneous confidence bounds.

*(English)*Zbl 0554.62029Consider 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.

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 |