Bias Correction with Jackknife, Bootstrap, and Taylor Series

We analyze the bias correction methods using jackknife, bootstrap, and Taylor series. We focus on the binomial model, and consider the problem of bias correction for estimating f(p), where f ∈ C[0, 1] is arbitrary. We characterize the supremum norm of the bias of general jackknife and bootstrap estimators for any continuous functions, and demonstrate the in deleted jackknife, different values of d may lead to drastically different behavior in jackknife. We show that in the binomial model, iterating the bootstrap bias correction infinitely many times may lead to divergence of bias and variance, and demonstrate that the bias properties of the bootstrap bias corrected estimator after r−1 rounds is exactly the same as that of the r-jackknife estimator if a bounded coefficients condition is satisfied.