Performance of balanced bootstrap resampling in distribution function and quantile problems (Q1118246)

It is shown that balanced resampling improves on ordinary uniform resampling when employed to estimate distribution functions or quantiles, but that the improvement is not by an order of magnitude as happens in problems such as bias estimation. Maximum improvement occurs at the centre of the distribution, where performance increases by 175 \%. The increase is only 16 \% at the 2.5 \% and 97.5 \% quantiles. In general the improvement is by the factor \(\{1-\phi^ 2/(1-\Phi)\Phi \}^{-1}\), where \(\Phi\) is the standard normal distribution function and \(\phi =\Phi '\).