Cross-validation methodologies have been widely used as a means of selecting tuning parameters in nonparametric statistical problems. In this paper we focus on a new method for improving the reliability of cross-validation. We implement this method in the context of the kernel density estimator, where one needs to select the bandwidth parameter so as to minimize L 2 risk. This method is a two-stage subsampling-extrapolation bandwidth selection procedure, which is realized by first evaluating the risk at a fictional sample size m ( m ≤ sample size n ) and then extrapolating the optimal bandwidth from m to n . This two-stage method can dramatically reduce the variability of the conventional unbiased cross-validation bandwidth selector. This simple first-order extrapolation estimator is equivalent to the rescaled “bagging-CV” bandwidth selector in Hall and Robinson (2009) if one sets the bootstrap size equal to the fictional sample size. However, our simplified expression for the risk estimator enables us to compute the aggregated risk without any bootstrapping. Furthermore, we developed a second-order extrapolation technique as an extension designed to improve the approximation of the true optimal bandwidth. To select the optimal choice of the fictional size m given a sample of size n , we propose a nested cross-validation methodology. Based on simulation study, the proposed new methods show promising performance across a wide selection of distributions. In addition, we also investigated the asymptotic properties of the proposed bandwidth selectors. 
Improving cross-validated bandwidth selection using subsampling-extrapolation techniques