BOOTSTRAP CALIBRATION FOR CONFIDENCE-INTERVAL CONSTRUCTION AND SELECTION

This article has three aims. First it is shown that, in simple situations, an exact procedure exists for implementing the calibration method of bootstrap interval construction. The procedure helps reveal the relationships between the calibration, bootstrap t and bootstrap root methods, and identifies settings in which all three methods yield the same result. Secondly, a method of bootstrap interval construction is introduced which has the nice property of requiring only one level of bootstrap resampling, but which yields coverage error rates that are smaller than those obtained with the bootstrap t method. The method depends on the calibration of an Edgeworth-corrected confidence set, and its justification rests on Edgeworth expansions. Coverage error rates of order O(n-3/2) and order O(n-2) or smaller are obtained for one-sided and two-sided intervals respectively. Because of the proliferation of numerous techniques for interval construction with different orders of asymptotic error rate, the practical problem of choosing among candidate intervals is becoming increasingly important. The third aim of this paper is to propose calibration as a method selection tool. It is shown that when the candidate intervals are derived from Edgeworth-corrected statistics, the calibration-selected interval possesses an asymptotic error rate equal to the best among them. Simulation results indicate that the finite-sample properties of the interval are also quite satisfactory.