A Note on Importance Resampling for Multi-Dimensional Statistics

Abstract

Johns (1988), Davison (1988), and Do and Hall (1991) used importance sampling for calculating bootstrap distributions of one-dimensional statistics. Realizing that their methods can not be extended easily to multi-dimensional statistics, Fuh and Hu (2004) proposed an exponential tilting formula for statistics of multi-dimension, which is optimal in the sense that the asymptotic variance is minimized for estimating tail probabilities of asymptotically normal statistics. For one-dimensional statistics, Hu and Su (2008) proposed a multi-step variance minimization approach that can be viewed as a generalization of the two-step variance minimization approach proposed by Do and Hall (1991). In this article, we generalize the approach of Hu and Su (2008) to multi-dimensional statistics, which applies to general statistics and does not resort to asymptotics. Empirical results on a real survival data set show that the proposed algorithm provides significant computational efficiency gains.

Keywords:

• Bootstrap
• Importance resampling
• Minimization
• Multi-dimensional
• Variance

Mathematics Subject Classification:

• 62G09
• 62G15
• 62G32
• 62N01

Acknowledgments

We thank the three anonymous reviewers for their insightful comments and suggestions, which have led to an improved article.