Refining Bootstrap Simultaneous Confidence Sets

Abstract

Abstract Simultaneous confidence sets for a collection of parametric functions may be constructed in several different ways. These ways include: (a) the exact pivotal method that underlies Tukey's (1953) and Scheffé's (1953) simultaneous confidence intervals for linear parametric functions in the normal linear model; (b) the method of asymptotic pivots, which is an approximate extension of the pivotal method; (c) the method of bootstrapped roots developed in Beran (1988). These three methods share several features. Each method simultaneously asserts a collection of confidence sets, one confidence set for every parametric function of interest. Each method obtains the uth constituent confidence set by referring a root to a critical value; the uth root is a function of the sample and of the uth parametric function. Each method has the same aim: to control the overall level of the simultaneous confidence set and to keep equal the marginal levels of the individual confidence statements that make up the simultaneous confidence set. The three methods differ in how they construct the necessary critical values. The pivotal method requires that the roots be identically distributed and that the supremum over all the roots be a pivot—a function of the sample and of the unknown parameter whose distribution is completely known. The method of asymptotic pivots relaxes these assumptions slightly by requiring them to hold only asymptotically. Nevertheless, the practical scope of both methods is very narrow. In contrast, the method of bootstrapped roots, or B method, is widely applicable. Under general conditions, it generates simultaneous confidence sets that, asymptotically, are balanced and have correct overall coverage probability. Moreover, the B method has a direct Monte Carlo approximation, which makes it easy to use. Under supplementary conditions, the pivotal method turns out to be a special case of the B method. At finite sample sizes, a B method simultaneous confidence set may suffer error in its overall coverage probability and lack of balance among the marginal coverage probabilities of its constituent confidence sets. Of what orders are these errors? What can be done to reduce them? We answer both questions in this article, by introducing and studying the B 2 method—a double iteration of the B method. Under regularity conditions, the B 2 method reduces the asymptotic order of imbalance in the B method simultaneous confidence set; and at the same time, it reduces the asymptotic order of error in overall coverage probability. The B 2 method has a direct Monte Carlo approximation that makes its use straightforward, though computer-intensive.