Extrinsic Semiparametric Methods for Spherical Regression: Models, Robust Estimation and Applications

Abstract

Spherical data are observations that lie on the unit sphere and are often represented as unit vectors. This geometric constraint poses challenges for the analysis of such data, making standard Euclidean regression models infeasible. This thesis studies regression models with spherical responses and Euclidean covariates. Existing methods for spherical regression, mainly parametric, often 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 in the regression setting, despite the common presence of outliers in highly concentrated directional data. To address this gap, Chapter 2 introduces a flexible and computationally efficient regression framework, the extrinsic single-index model (ESIM). ESIM combines a nonparametric link function with interpretable covariate coefficients, which enables both model flexibility and interpretability. We propose simultaneous M-estimators for both components of ESIM, and achieve robust estimation using a suitable choice of loss function. The large-sample properties of the estimators are established, and a Wald-type statistic is developed for robust inference on the parametric component. Chapter 3 provides theoretical justification of the robustness properties of ESIM. Robustness is assessed via the influence function and standardized influence function. The principal focus is on the case where the spherical responses have error distributions which are highly concentrated and have elliptical symmetry around the mean direction. These features are commonly observed in real directional data but have not been adequately addressed in the regression setting Special attention is given to the exponential squared loss (ESL), which offers comparable efficiency and superior robustness, compared with the least squares loss, in this setting. We also examine how to choose the tuning parameter for the ESL, so as to balance efficiency and robustness. We provide theoretical guidance on the optimal choice of the ESL tuning parameter. Extensive numerical studies in Chapter 4 substantiate the computational efficiency and robustness of our methods. Show more