Higher-Order Asymptotic Properties of Kernel Density Estimator with Global Plug-In and Its Accompanying Pilot Bandwidth

This study investigates the effect of bandwidth selection via a plug-in method on the asymptotic structure of the nonparametric kernel density estimator. We generalise the result of Hall and Kang (2001) and find that the plug-in method has no effect on the asymptotic structure of the estimator up to the order of $O\{(nh_0)^{-1/2}+h_0^L\}=O(n^{-L/(2L+1)})$ for a bandwidth $h_0$ and any kernel order $L$ when the kernel order for pilot estimation $L_p$ is high enough. We also provide the valid Edgeworth expansion up to the order of $O\{(nh_0)^{-1}+h_0^{2L}\}$ and find that, as long as the $L_p$ is high enough , the plug-in method has an effect from on the term whose convergence rate is $O\{(nh_0)^{-1/2}h_0+h_0^{L+1}\}=O(n^{-(L+1)/(2L+1)})$. In other words, we derive the exact achievable convergence rate of the deviation between the distribution functions of the estimator with a deterministic bandwidth and with the plug-in bandwidth. In addition, we weaken the conditions on kernel order $L_p$ for pilot estimation by considering the effect of pilot bandwidth associated with the plug-in bandwidth. We also show that the bandwidth selection via the global plug-in method possibly has an effect on the asymptotic structure even up to the order of $O\{(nh_0)^{-1/2}+h_0^L\}$. Finally, Monte Carlo experiments are conducted to see whether our approximation improves previous results.