AI 中文总结
针对高维核密度估计的维度灾难，提出基于遗传算法选择变量子集的混合密度估计器，通过平衡近似误差与方差缩减，在权衡适当时性能优于传统全维估计器。
AI 中文摘要
高维核密度估计面临维度灾难问题。本研究提出一种混合密度估计器，其定义为预先指定维度子集上的联合密度与其余变量的边缘密度的乘积，而非直接估计完整的高维密度。在该框架下，给定观测数据，我们选择一个变量子集用于构建联合密度以提升估计精度，同时通过各自的边缘密度对其余变量建模。该构造涉及简化依赖结构所带来的近似误差，与通过降低联合密度维度实现的方差缩减之间的权衡。我们采用遗传算法高效识别此类变量子集。理论与数值结果表明，当这种权衡得到适当平衡时，所提出的估计器可优于传统的全维核密度估计。
英文摘要
High-dimensional kernel density estimation suffers from the curse of dimensionality. This study proposes a hybrid density estimator defined as the product of a joint density over a pre-specified subset of dimensions and marginal densities for the remaining variables, instead of estimating the full high-dimensional density directly. Under this framework, given the observed data, we select a subset of variables for joint density construction to improve estimation accuracy, while modeling the remaining variables through their respective marginal densities. This construction involves a trade-off between the approximation error induced by simplifying the dependence structure and the variance reduction achieved by lowering the dimension of the joint density. We employ a genetic algorithm to efficiently identify such variable subsets. Theoretical and numerical results demonstrate that the proposed estimator can outperform conventional full-dimensional kernel density estimation when this trade-off is appropriately balanced.
