EXACT MEAN INTEGRATED SQUARED ERROR OF HIGHER ORDER KERNEL ESTIMATORS
推导了非参数核密度估计中高阶核的精确均方积分误差，对比了三种核的有限样本效率，发现核阶数选择比核类型更重要，并提出了基于最小化最大遗憾的核阶数选择方法。
The exact mean integrated squared error (MISE) of the nonparametric kernel density estimator is derived for the asymptotically optimal smooth polynomial kernels of Müller (1984, Annals of Statistics 12, 766–774) and the trapezoid kernel of Politis and Romano (1999, Journal of Multivariate Analysis 68, 1–25) and is used to contrast their finite-sample efficiency with the higher order Gaussian kernels of Wand and Schucany (1990Canadian Journal of Statistics 18, 197–204). We find that these three kernels have similar finite-sample efficiency. Of greater importance is the choice of kernel order, as we find that kernel order can have a major impact on finite-sample MISE, even in small samples, but the optimal kernel order depends on the unknown density function. We propose selecting the kernel order by the criterion of minimax regret, where the regret (the loss relative to the infeasible optimum) is maximized over the class of two-component mixture-normal density functions. This minimax regret rule produces a kernel that is a function of sample size only and uniformly bounds the regret below 12% over this density class.The paper also provides new analytic results for the smooth polynomial kernels, including their characteristic function.This research was supported in part by the National Science Foundation. I thank Oliver Linton and a referee for helpful comments and suggestions that improved the paper.