Balanced Simultaneous Confidence Sets

Abstract

Abstract Suppose that T(θ) = {Tu (θ)} is a family of parametric functions indexed by the variable u. For each u, an approximate confidence set Cn,u for Tu (θ) may be obtained by referring a function of Tu (θ) and the sample to an estimated quantile dn,u of that function's sampling distribution. This approach is sometimes called the pivotal method for constructing a confidence set, even when it is not based on a true pivot. A simultaneous confidence set Cn for T(θ) is then obtained by simultaneously asserting the individual confidence sets {Cn,u }. The problem is to choose the critical values for the {Cn,u } in such a way that the overall coverage probability of Cn is correct and the coverage probabilities of the individual confidence sets {Cn,u } are equal. The second property is termed balance. It means that the simultaneous confidence set Cn treats each constituent confidence statement Cn,u fairly. Aside from a few special cases, the problem just described is too difficult for analytical approaches, whether exact or asymptotic in nature. A new bootstrap method described in this article, however, yields a practical solution to the problem that is asymptotically valid under general conditions. The new method recovers, as special cases, the Tukey and Scheffé simultaneous confidence intervals for the normal linear model. In addition, it handles distributionally harder problems, such as simultaneous confidence cones for the eigenvectors of an unknown covariance matrix or confidence bands for a linear predictor, in either parametric or nonparametric settings.