Practical higher-order smoothing of the bootstrap

In the context of functional estimation, the bootstrap approach amounts to substitution of the empirical distribution function for the unknown underlying distribution in the definition of the functional. A smoothed bootstrap alternative substitutes instead a smoothed version of the empirical distribution function, obtained by kernel smoothing of the given data sample. It may be theoretically advantageous to base such a smoothed bootstrap estimator on a higher-order kernel density estimator.Such density estimators necessarily take negative values, which creates a practical problem when simulation is to be used in construction of the bootstrap estimator. We illustrate how a negativity correction may be combined with rejection samplingto make higher-order smoothing feasible in the bootstrap context. Estimation of the variance of a sample quantile is examined both theoretically and in a simulation study.