A General Theory for Jackknife Variance Estimation

The delete-1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles. This well-known deficiency can be rectified by using a more general jackknife with d, the number of observations deleted, depending on a smoothness measure of the point estimator. Our general theory explains why jackknife works or fails. It also shows that (i) for "sufficiently smooth" estimators, the jackknife variance estimators with bounded d are consistent and asymptotically unbiased and (ii) for "nonsmooth" estimators, d has to go to infinity at a rate explicitly determined by a smoothness measure to ensure consistency and asymptotic unbiasedness. Improved results are obtained for several classes of estimators. In particular, for the sample p-quantiles, the jackknife variance estimators with d satisfying n1/2/d → 0 and n - d → ∞ are consistent and asymptotically unbiased.