Computation of Exact Bootstrap Confidence Intervals: Complexity and Deterministic Algorithms

The bootstrap is a nonparametric approach for calculating quantities, such as confidence intervals, directly from data. Since calculating exact bootstrap quantities is believed to be intractable, randomized resampling algorithms are traditionally used. In this paper, we present a new perspective on the bootstrapmethod through the lens of counting integer points in polyhedra. Through this new perspective, we make several advances for the bootstrap method, both theoretically and algorithmically. First, we establish several computational complexity results for the exact bootstrapmethod in the case of the samplemean. Second, we present the first efficient deterministic approximation algorithm (fully polynomial time approximation scheme) for producing exact bootstrap confidence intervals which, unlike traditional methods, has guaranteed bounds on the approximation error. Third, we develop a simple exact algorithm for exact bootstrap confidence intervals based on polynomial multiplication. We provide empirical evidence on real and synthetic data sets with several hundreds of data points that the proposed deterministic algorithms can quickly produce confidence intervals that are substantially more accurate than those from randomized methods, and thus are practical alternatives in applications such as clinical trials.