Jackknife-After-Bootstrap Standard Errors and Influence Functions

SUMMARY This paper shows how to derive more information from a bootstrap analysis, information about the accuracy of the usual bootstrap estimates. Suppose that we observe data x = (x 1 x 2, . . ., xn), compute a statistic of interest s(x) and further compute B bootstrap replications of s, say s(x*1) s(x*2), . . ., s(x*B), where B is some large number like 1000. Various accuracy measures for s(x) can be obtained from the bootstrap values, e.g. the bootstrap estimates of standard error and bias, or the length and shape of bootstrap confidence intervals. We might wonder how accurate these accuracy measures themselves are, or how sensitive they are to small changes in the individual data points xi. It turns out that these questions can be answered from the information in the original bootstrap sample s* 1 s* 2, . . ., s*B, with no further resampling required. The answers, which make use of the jackknife and delta method influence functions, are easy to apply and can give informative results, as shown by several examples.