Regression with a spherical response is challenging due to the absence of linear structure, making standard regression models inadequate. Existing methods, mainly parametric, lack the flexibility to capture the complex relationship induced by spherical curvature, while methods based on techniques from Riemannian geometry often suffer from computational difficulties. The non-Euclidean structure further complicates robust estimation, with very limited work addressing this issue, despite the common presence of outliers in directional data. This article introduces a new semi-parametric approach, the extrinsic single-index model (ESIM) and its robust estimation, to address these limitations. We establish large-sample properties of the proposed estimator with a wide range of loss functions and assess their robustness using the influence function and standardized influence function. Specifically, we focus on the robustness of the exponential squared loss (ESL), demonstrating comparable efficiency and superior robustness over least squares loss under high concentration. We also examine how the tuning parameter for the ESL balances efficiency and robustness, providing guidance on its optimal choice. The computational efficiency and robustness of our methods are further illustrated via simulations and applications to geochemical compositional data.

翻译：球面响应变量的回归分析因缺乏线性结构而具有挑战性，使得标准回归模型不再适用。现有方法主要为参数化方法，缺乏捕捉球面曲率所诱导复杂关系的灵活性；而基于黎曼几何技术的方法则常受计算困难所困扰。非欧几里得结构进一步使稳健估计复杂化，尽管方向性数据中普遍存在异常值，但针对此问题的研究极为有限。本文提出一种新的半参数方法——外征单指标模型（ESIM）及其稳健估计，以应对这些局限。我们建立了所提估计量在广泛损失函数下的大样本性质，并使用影响函数及标准化影响函数评估其稳健性。具体而言，我们聚焦于指数平方损失（ESL）的稳健性，证明在高浓度条件下其具有与最小二乘损失相当的有效性及更优的稳健性。我们还研究了ESL调节参数如何平衡有效性与稳健性，为其最优选择提供指导。通过模拟实验及对地球化学成分数据的应用，进一步展示了所提方法的计算效率与稳健性。