AI 中文总结
研究针对一般黎曼流形上的预测器及一般度量空间的响应，提出局部常数和局部线性估计器，是首个此设置下的局部线性弗雷歇回归方法，构建中采用多种技术，通过模拟和实际应用验证了方法性能。
AI 中文摘要
弗雷歇回归在欧几里得预测器方面已得到充分发展，但局部线性方法在一般流形值预测器上仍有局限。我们针对位于一般黎曼流形上的预测器和取值于一般度量空间的响应，提出了局部常数和局部线性估计器。所提局部线性估计器是此设置下的首个局部线性弗雷歇回归方法。我们的构建使用测地邻域、对数映射坐标、体积密度校正和框架不变标量等效权重。对于这两种估计器，我们不仅建立了逐点一致性和收敛速率，还建立了一致一致性和收敛速率。模拟和实际数据应用证明了所提方法在不同预测器和响应几何结构中的有限样本性能和实际适用性。
英文摘要
Fréchet regression is well developed for Euclidean predictors, but local linear methods remain limited for general manifold-valued predictors. We propose local constant and local linear estimators for predictors lying on a general Riemannian manifold and responses taking values in a general metric space. The proposed local linear estimator is the first local linear Fréchet regression method in this setting. Our construction uses geodesic neighborhoods, logarithmic-map coordinates, volume-density correction, and frame-invariant scalar equivalent weights. For both estimators, we establish not only pointwise consistency and convergence rates but also uniform consistency and convergence rates. Simulations and real data applications demonstrate the finite-sample performance and practical applicability of the proposed methods across diverse predictor and response geometries.
Comments178 pages, 4 figures, 3 tables
