A1708
Title: Optimal geodesic regression for directional response
Authors: Gaspard Bernard - Academia Sinica, Institute of Statistical Science (Taiwan) [presenting]
Abstract: A directional response variable Y taking values on the unit sphere in dimension 3 is considered, together with linear covariates X. The problem of estimating a regression function valued on the unit sphere has received considerable attention. However, most of the existing work focuses on nonparametric methods, for which both interpretability and optimality may be difficult to achieve. A single-index framework is considered in which the regression function is assumed to be a geodesic curve and the noise is assumed to lie in the tangent space. This model has been introduced in the context of general Riemannian manifolds, but the aim is to propose a natural analogue for directional responses of the classical linear model. Under the assumption that the tangent space noise distribution is spherically symmetric with specified radial density, this approach enables the derivation of locally and asymptotically optimal tests and confidence intervals for both the geodesic curve itself and the coefficients of the single index. The potential asymptotic cost of performing inference on only one of these two parameters when the other is treated as a nuisance is then investigated. Subsequently, the question of how to preserve the validity of the proposed tests when the radial density of the noise is unspecified while retaining their optimality properties is studied. Finally, the extension of the proposed framework to responses taking values on hyperspheres of arbitrary dimension is discussed.
