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This study investigates the effect of bandwidth selection via a plug-inmethod on the asymptotic structure of the nonparametric kernel densityestimator. We generalise the result of Hall and Kang (2001) and find that theplug-in method has no effect on the asymptotic structure of the estimator up tothe 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 $L_p$ ishigh 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, wederive the exact achievable convergence rate of the deviation between thedistribution functions of the estimator with a deterministic bandwidth and withthe plug-in bandwidth. In addition, we weaken the conditions on kernel order$L_p$ for pilot estimation by considering the effect of pilot bandwidthassociated with the plug-in bandwidth. We also show that the bandwidthselection via the global plug-in method possibly has an effect on theasymptotic 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 approximationimproves previous results.

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