Higher-Order Coverage Errors of Batching Methods via Edgeworth Expansions on $t$-Statistics

Abstract

While batching methods have been widely used in simulation and statistics, it is open regarding their higher-order coverage behaviors and whether one variant is better than the others in this regard. We develop techniques to obtain higher-order coverage errors for batching methods by building Edgeworth-type expansions on t-statistics. The coefficients in these expansions are intricate analytically, but we provide algorithms to estimate the coefficients of the n −1 error term via Monte Carlo simulation. We provide insights on the effect of the number of batches on the coverage error where we demonstrate generally non-monotonic relations. We also compare different batching methods both theoretically and numerically, and argue that none of the methods is uniformly better than the others in terms of coverage. However, when the number of batches is large, sectioned jackknife has the best coverage among all.

A Cheap Bootstrap Method for Fast Inference

“…However, unlike these existing methods, the pivotal statistic used in Cheap Bootstrap has a limiting t-distribution, not a normal distribution. Edgeworth expansions on limiting t-distributions appear open to our best knowledge (except a recent working paper He and Lam (2021)). Here, we derive our expansions for Cheap Bootstrap by integrating the expansions of the original estimate and resample estimates that follow (conditional) asymptotic normal distributions.…”

“…Different from conventional bootstrap procedures, the Cheap Bootstrap pivotal statistic has a limiting t-distribution, not a normal distribution. Edgeworth expansion on t-distribution is, to our best knowledge, open in the literature (except the recent working paper He and Lam (2021)). Here, we build the Edgeworth expansions for Cheap Bootstrap coverage probabilities and show their errors match the order O(n −1 ) (where n is the sample size) incurred in conventional bootstraps in the two-sided case.…”