Smoothed pseudo-population bootstrap methods with applications to finite population quantiles

Abstract This article introduces smoothed pseudo-population bootstrap methods for the purposes of mean-squared error estimation and for constructing confidence intervals for finite population quantiles. In an independent and identically distributed context, it has been shown that resampling from a smoothed estimate of the distribution function instead of the usual empirical distribution function can improve the convergence rate of the bootstrap mean-squared error estimator of a sample quantile. We extend the smoothed bootstrap to the survey sampling framework by implementing it in pseudo-population bootstrap methods for high entropy, single-stage survey designs, such as simple random sampling without replacement, Poisson sampling, and randomized systematic proportional-to-size sampling. Given a kernel function and a bandwidth, it consists of smoothing the pseudo-population from which bootstrap samples are drawn using the original sampling design. Given that the implementation of the proposed algorithms requires the specification of the bandwidth, we develop a plug-in selection method along with a grid search selection method based on a bootstrap estimate of the mean squared error. Simulation results suggest that the smoothed approach offers improved efficiency compared to the standard pseudo-population bootstrap for estimating the uncertainty of a quantile estimator, together with mixed results regarding confidence interval coverage.