Bias Correction with Jackknife, Bootstrap, and Taylor Series
This work addresses bias estimation problems for statisticians, providing theoretical insights into the limitations of common correction methods, but it is incremental as it builds on existing jackknife and bootstrap techniques.
The paper tackles bias correction in the binomial model for estimating arbitrary continuous functions, showing that different delete-d jackknife values lead to varied behaviors and that infinite bootstrap iterations can cause divergence in bias and variance.
We analyze 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 $f \in C[0,1]$ is arbitrary. We characterize the supremum norm of the bias of general jackknife and bootstrap estimators for any continuous functions, and demonstrate the in delete-$d$ jackknife, different values of $d$ may lead to drastically different behaviors 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 are of the same order as that of the $r$-jackknife estimator if a bounded coefficients condition is satisfied.