Jackknife Approximations to Bootstrap Estimates

Let Tₙ be an estimate of the form Tₙ = T (Fₙ), where Fₙ is the 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 results: the first-order Edgeworth expansion estimate for the distribution of n^1/2 (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).