An exact bootstrap-based bandwidth selection rule for kernel quantile estimators

Abstract

Bandwidth selection is key for kernel quantile estimators (KQEs), which estimate quantiles by averaging all of the order statistics with appropriate kernel-weighting functions. This paper provides a new data-driven bandwidth selection method for KQEs, named the exact bootstrap-based bandwidth selection (EBS) rule. By relying on the exact analytic expressions for the bootstrap mean and variance of KQEs, the error due to bootstrap resampling is eliminated, and thus the optimal bandwidth can be obtained by minimizing the mean squared error (MSE) estimate. The effectiveness of this EBS rule is confirmed by numerical experiments. First, the bandwidth selection performance of the EBS method is compared to that of a benchmark approach. Simulation studies show that the EBS method performs well, especially in selecting bandwidths for extreme quantiles and when applied to small sample sizes with skewed distributions and relatively large variances. Second, KQEs with a bandwidth determined by our EBS rule is compared with five other state-of-the-art quantile estimators over six typical distributions. The results further validate the efficiency of the EBS method. Third, the results of simulations of controlling actual type-I error rates that occur when two independent groups are compared through quantiles further demonstrate the precision of our EBS-based KQEs.