Fast estimators of the jackknife

The jackknife is a reliable method for estimating standard error nonparametrically. The method is easy to use, but is computationally intensive. The time required to compute the jackknife standard error for an estimator <(theta)over cap> will depend on the time required to compute <(theta)over cap> itself. For some estimators the required time is prohibitive. This may be especially true in simulation studies where <(theta)over cap> and its standard error are computed for a large number of datasets. Let X-1, X-2, ..., X-N be a random sample and <(theta)over cap>((i)) the estimator computed with X-i removed. Then <(sigma)over cap>(2)(JK)/N = ([N-1]/N) Sigma(i=1)(N) (<(theta)over cap>((i))-<(theta)over cap>((.,N)))(2) is the jackknife estimator of the variability of <(theta)over cap> where <(theta)over cap>((.,N)) = (1/N) Sigma(i=1)(N) <(theta)over cap>((i)). In this paper estimators of <(sigma)over cap>(2)(JK), are defined that can be computed quickly while sacrificing little precision or accuracy. The method requires that random variables {<(theta)over tilde>((i))}(i=1)(N) are available that can be computed quickly and are strongly correlated with {<(theta)over cap>((i))}(i=1)(N). It is described how {<(theta)over tilde>((i))}(i=1)(N) can generally be obtained, and the method is illustrated with two examples. The paper focuses on the jackknife estimator for standard error, but the method can also be applied to quickly compute the jackknife estimator of bias.