AI 中文总结
本文提出无需验证一致等度连续性的新框架，证明多种Fréchet回归模型满足该条件，将加权Fréchet聚合和超限集估计框架扩展至非欧几里得空间，经模拟和实际应用验证其理论性质。
AI 中文摘要
非欧几里得空间中面向对象数据的统计分析严重依赖广义条件Fréchet均值，尤其在Fréchet回归场景中。但建立这些估计量的一致收敛性面临若干理论挑战，困难主要源于一般度量空间缺乏线性结构，使得验证估计量渐近一致等度连续性的标准技术难以应用。为克服这一局限，本文提出一种新的理论框架以建立一致收敛性，该框架无需验证一致等度连续性，仅需广义条件Fréchet均值的经验损失函数满足一种新的结构条件。我们证明，在广泛类别的度量空间中，多种知名的Fréchet回归模型均满足该分析条件。基于这些基础性的一致收敛保证，我们随后将两个广泛使用的框架从欧几里得空间扩展至非欧几里得空间：（i）加权Fréchet聚合框架，可实现分布式回归和稳健均值中位数回归；（ii）超限集估计框架，用于识别条件广义Fréchet均值超过规定阈值的关键协变量区域，同时提供量化超限总量的度量。通过蒙特卡洛模拟和对纽约市动态交通网络的应用，我们实证验证了这些所提方法的理论性质。
英文摘要
The statistical analysis of object oriented data in non-Euclidean spaces heavily relies on generalized conditional Fréchet means, notably in the context of Fréchet regression. However, establishing the uniform convergence of these estimators presents several theoretical challenges. The difficulties are caused primarily by the absence of linear structures in general metric spaces, rendering standard techniques for verifying the asymptotic uniform equicontinuity of the estimator largely intractable. To overcome this limitation, this paper introduces an alternative theoretical framework for establishing uniform convergence that bypasses the need to verify uniform equicontinuity, under a novel structural condition on the empirical cost function of the generalized conditional Fréchet means. We demonstrate that this analytical condition is satisfied by various prominent Fréchet regression models across broad classes of metric spaces. Leveraging these foundational uniform convergence guarantees, we subsequently extend two widely used frameworks from Euclidean to non-Euclidean spaces: (i) a weighted Fréchet aggregation framework that facilitates both distributed regression and robust median-of-means regression; and (ii) an exceedance set estimation framework to identify critical covariate regions where the conditional generalized Fréchet mean surpasses a prescribed threshold, alongside a metric to quantify the aggregate magnitude of the exceedance. The theoretical properties of these proposed methods are empirically validated through Monte Carlo simulations and an application to dynamic transportation networks in New York City.
