  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.
