Multivariate Boundary Kernels and a Continuous Least Squares Principle
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多变量边界核与连续最小二乘原理
- 研究问题
- 对于具有复杂边界的紧支撑函数，如何构造多变量边界核，使密度和曲线的核估计具有整体一致性？
- 方法与思路
- 作者在各边界区域估计点求解核生成变分问题，将内部核自然延续至边界，并讨论数值实现及局部加权最小二乘联系。
- 材料陈述的贡献
- 摘要称，所构造边界核可处理任意支撑形状，并建立连续最小二乘原理和 Gauss–Markov 定理的连续类比。
- 阅读关注点
- 应核对支撑集与边界的实际条件、变分问题和核的约束，以及积分均方误差和上确界范数下一致性结论的适用范围。
- 阅读边界
- 仅据所给 Crossref 摘要，不能将任意支撑的表述理解为无需其他条件；摘要未给出正则条件、收敛率或数值实现成本。
原文摘要
Summary Whereas there are many references on univariate boundary kernels, the construction of boundary kernels for multivariate density and curve estimation has not been investigated in detail. The use of multivariate boundary kernels ensures global consistency of multivariate kernel estimates as measured by the integrated mean-squared error or sup-norm deviation for functions with compact support. We develop a class of boundary kernels which work for any support, regardless of the complexity of its boundary. Our construction yields a boundary kernel for each point in the boundary region where the function is to be estimated. These boundary kernels provide a natural continuation of non-negative kernels used in the interior onto the boundary. They are obtained as solutions of the same kernel-generating variational problem which also produces the kernel function used in the interior as its solution. We discuss the numerical implementation of the proposed boundary kernels and their relationship to locally weighted least squares. Along the way we establish a continuous least squares principle and a continuous analogue of the Gauss–Markov theorem.