Importance Resampling for Bootstrap Confidence Regions by Cheng-der Fuh And

In this article, we obtain an importance resampling formula to reduce the amount of resampling necessary for the construction of bootstrap con dence regions. In the one-dimensional case, the formula reduces to that of Jones (1988) and Do & Hall (1991). However, the methods employed by previous authors are not tamed for direct generalization to multi-dimensional parameters. Therefore no formula is available for bootstrap con dence regions in the literature. Our method, which is closely related to the large deviation tilting, allows a relatively easy treatment of the multi-dimensional case. The method also reveals a phenomenon that happens only in the multi-dimensional case. That is, the optimally tilted distribution is a mixture of exponentially tilted distributions, and the mixture components depend on the shape of the con dence region. We also provide a general account of the importance resampling in relation to the large deviation tilting. E ciency properties of the simulation scheme using the proposed formula is established together with numerical evidence. Some key words: Bootstrap; Con dence region; Importance resampling; Large deviation; Uniform resampling Research partially supported by the National Science Council of ROC. Visiting National Taiwan University and Academia Sinica, research partially supported by Hong Kong Research Grant Council.