Bootstrap variance estimation for Nadaraya quantile estimator

Under a suitable choice of bandwidth, Nadaraya's estimator of the pth quantile yields smaller mean squared error than the unsmoothed pth sample quantile. We investigate the problem of bootstrap estimation of the variance of the Nadaraya quantile estimator and show that the error of the variance estimator can be reduced by smoothing the bootstrap. A novel approach, which calibrates the order p of the bootstrapped Nadaraya quantile estimates, is shown to reduce the error further. A simulation study is reported on the empirical performance of the proposed modified bootstrap variance estimators.