Abstract: 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 sampling to 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.
Key words and phrases: Kernel function, negativity correction, rejection sampling, sample quantile.