标题： 间接交叉验证用于密度估计 显示英文标题

标题： Indirect Cross-validation for Density Estimation

摘要： 提出了一种用于核密度估计器的带宽选择新方法。该方法称为间接交叉验证（ICV），利用所谓的选择核。最小二乘交叉验证（LSCV）被用来选择选择核估计量的带宽，并且该带宽经过适当缩放后可用于高斯核估计量。所提出的选核是两个高斯核的线性组合，不必是单峰或正值。理论表明，ICV 带宽的相对误差可以以速率$n^{-1/4}$收敛到 0，这比 LSCV 的$n^{-1/10}$收敛率要好得多。有趣的是，对于带宽选择而言最佳的选择核如果用于实际估计密度函数，则表现非常差。这一特性似乎是更大且已充分记录的悖论的一部分，即“估计问题越难，交叉验证的表现越好。” 在模拟研究、真实数据实例以及局部选择带宽的模拟例子中，ICV 方法始终优于 LSCV。 显示英文摘要

摘要： A new method of bandwidth selection for kernel density estimators is proposed. The method, termed indirect cross-validation, or ICV, makes use of so-called selection kernels. Least squares cross-validation (LSCV) is used to select the bandwidth of a selection-kernel estimator, and this bandwidth is appropriately rescaled for use in a Gaussian kernel estimator. The proposed selection kernels are linear combinations of two Gaussian kernels, and need not be unimodal or positive. Theory is developed showing that the relative error of ICV bandwidths can converge to 0 at a rate of $n^{-1/4}$, which is substantially better than the $n^{-1/10}$ rate of LSCV. Interestingly, the selection kernels that are best for purposes of bandwidth selection are very poor if used to actually estimate the density function. This property appears to be part of the larger and well-documented paradox to the effect that "the harder the estimation problem, the better cross-validation performs." The ICV method uniformly outperforms LSCV in a simulation study, a real data example, and a simulated example in which bandwidths are chosen locally.