Jackknife Approximations to Bootstrap Estimates

Let T be an estimate of the form Tn = T(F ) where F is the nn n' n sample cdf of n iid observations and T is a locally quadratic functional defined on cdf's. Then, the normalized jackknife estimates for bias, skewness, and variance of Tn approximate closely their bootstrap counterparts. Each of these estimates is consistent. Moreover, the jackknife and bootstrap estimates of variance are asymptotically normal and asymptotically minimax. The main result: the first-order Edgeworth expansion estimate for the distribution of n1T2(Tn-T(F)), with F being the actual cdf of each observation and the expansion coefficients being estimated by jackknifing, is asymptotically equivalent to the corresponding bootstrap distribution estimate, up to and including terms of order n 1/2 Both distribution estimates are asymptotically minimax. The jackknife Edgeworth expansion estimate suggests useful corrections for skewness and bias to upper and lower confidence bounds for T(F). AMS 1980 subJect classification: Primary 62G05; Secondary 62E20