论文标题
内核密度估计器的高阶渐近性能及其随附的飞行员带宽
Higher-Order Asymptotic Properties of Kernel Density Estimator with Global Plug-In and Its Accompanying Pilot Bandwidth
论文作者
论文摘要
这项研究通过插入方法研究带宽选择的影响对非参数核密度估计器的渐近结构。我们概括了Hall and Kang（2001）的结果，并发现插件方法对估计器的渐近结构没有影响，直到$ o \ {（nh_0）^{ - 1/2}+h_0^l \} = o（n^{ - l/（l/（2l+1））$ $ kern $ kern $ kern $ kern nh_0^l \} = o（飞行员估计$ L_P $足够高。我们还提供有效的Edgeworth扩展到$ o \ {（nh_0）^{ - 1}+H_0^{2l} {2l} \} $的顺序$ o \ {（nh_0）^{ - 1/2} h_0+h_0^{l+1} \} = o（n^{ - （l+1）/（2l+1）}）$。换句话说，我们通过确定性带宽和插件带宽得出了估算器分布函数之间偏差的确切可达到的收敛速率。此外，我们通过考虑与插件带宽相关的飞行器带宽的效果，削弱了内核订单$ L_P $的条件。我们还表明，通过全局插件方法的带宽选择可能会影响渐近结构，甚至达到$ o \ {（nh_0）^{ - 1/2}+H_0^l \} $的顺序。最后，进行了蒙特卡洛实验，以查看我们的近似是否改善了先前的结果。
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.
