Higher-Order Asymptotic Properties of Kernel Density Estimator with Global Plug-In and Its Accompanying Pilot Bandwidth
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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)^-1/2h_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.
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