多项式相依下的空间函数 $k$-最近邻回归
Spatial Functional $k$-Nearest-Neighbour Regression under Polynomial Dependence
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中文总结 AI 辅助
本文针对空间格点上的函数型数据，提出并分析了 $k$-最近邻回归估计量，通过夹逼框架解耦小球轮廓与协方差衰减，推导出不同空间机制下的收敛速率，并验证了理论结果。
中文摘要 AI 辅助
本文研究了当解释变量取值于可分希尔伯特空间且观测值在递增的规则空间格点上采样时的非参数回归估计。在协变量与误差之间严格的场对场独立性设定下，我们探讨了函数 $k$-最近邻（$k$-NN）估计量的结构性和渐近集中性质。通过为数据驱动的随机带宽建立尖锐的逐路径确定性夹逼框架，我们成功地将局部无限维小球轮廓与邻域指标的多项式协方差衰减解耦。在短程、临界和长程空间机制下推导了点态收敛速率，揭示了一个反映协变量空间交互和响应误差记忆的组合惩罚项。此外，对于无界次高斯误差过程，通过度量熵和指标边界壳链式论证，建立了紧子集上的一致一致性。结构化多项式模拟证实了我们的理论速率，并展示了空间长程瓶颈的精确机制。
英文摘要
This paper investigates non-parametric regression estimation when the explanatory variable takes values in a separable Hilbert space and observations are sampled over an increasing regular spatial lattice. Under a rigorous field-to-field independence setup between covariates and errors, we explore the structural and asymptotic concentration properties of the functional $k$-nearest-neighbour ($k$-NN) estimator. By establishing a sharp pathwise deterministic sandwiching framework for the data-driven random bandwidth, we successfully decouple the local infinite-dimensional small-ball profile from the polynomial covariance decay of the neighborhood indicators. Pointwise convergence rates are derived across short-range, critical, and long-range spatial regimes, revealing a combined penalty term that reflects both covariate spatial interaction and response error memory. Furthermore, uniform consistency over compact subsets is established for unbounded sub-Gaussian error processes through metric entropy and indicator boundary shell chaining. Structured polynomial simulations confirm our theoretical rates and exemplify the precise mechanics of the spatial long-range bottleneck.
发表机构
- Université des Sciences et Techniques de Masuku (USTM)（马苏库科学技术大学）
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