Higher-Order Asymptotic Properties of Kernel Density Estimator with Plug-In Bandwidth

Abstract

This study investigates the effect of bandwidth selection via plug-in method on the asymptotic structure of the nonparametric kernel density estimator. We find that the plug-in method has no effect on the asymptotic structure of the estimator up to the order of O{(nh₀)⁻¹/²} = O(n[-L/(2L+1)]) for a bandwidth h0 and any kernel order L. We also provide the valid Edgeworth expansion up to the order of O{(nh₀)⁻¹} and find that the plug-in method starts to have an effect from on the term whose convergence rate is O{(nh₀)⁻¹/²h₀} = O(n[−(L+1)/(2L+1)]). In other words, we derive the exact convergence rate of the deviation between the distribution functions of the estimator with a deterministic bandwidth and with the plug-in bandwidth. Monte Carlo experiments are conducted to see whether our approximation improves previous results.