Sharpening estimators using resampling

We study ways to improve a given estimator using resampling methods like the jackknife or the bootstrap in terms of bias and the mean square error. Our key task is to devise a method to empirically check whether the bias correction employed leads to an increase or decrease in the mean square error in terms of second-order asymptotics. We derive conditions under which we can sharpen the given estimator in terms of bias and the mean square error. One may attempt to verify the condition empirically using resampling methods.