KERNEL ESTIMATION WHEN DENSITY MAY NOT EXIST
研究了密度函数不光滑或存在奇异成分时，核密度估计和条件均值估计的渐近性质，通过广义函数和广义随机过程方法给出了统一理论。
Nonparametric kernel estimation of density and conditional mean is widely used, but many of the pointwise and global asymptotic results for the estimators are not available unless the density is continuous and appropriately smooth; in kernel estimation for discrete-continuous cases smoothness is required for the continuous variables. Nonsmooth density and mass points in distributions arise in various situations that are examined in empirical studies; some examples and explanations are discussed in the paper. Generally, any distribution function consists of absolutely continuous, discrete, and singular components, but only a few special cases of nonparametric estimation involving singularity have been examined in the literature, and asymptotic theory under the general setup has not been developed. In this paper the asymptotic process for the kernel estimator is examined by means of the generalized functions and generalized random processes approach; it provides a unified theory because density and its derivatives can be defined as generalized functions for any distribution, including cases with singular components. The limit process for the kernel estimator of density is fully characterized in terms of a generalized Gaussian process. Asymptotic results for the Nadaraya–Watson conditional mean estimator are also provided.