A0771
Title: Empirical likelihood on manifolds
Authors: Andrew Wood - Australian National University (Australia) [presenting]
Karthik Bharath - University of Nottingham (United Kingdom)
Huiling Le - University of Nottingham (United Kingdom)
Abstract: Empirical likelihood (EL) is a type of nonparametric likelihood that is useful, e.g. for constructing nonparametric confidence regions in one-sample problems for intrinsic or extrinsic Frechet means and in k-sample testing problems for Frechet means, especially when one wishes to avoid the assumption of a common dispersion structure across populations. An important property of EL is that it obeys Wilk's theorem. EL has previously been developed for data on particular manifolds, including the unit sphere, with applications to directional statistics; real projective space, with applications to axial data; and complex projective space, with applications to (Kendall) shape data for objects represented by labelled landmarks in two dimensions. However, in previous work on EL, there has been no attempt to develop a general, unified approach to EL for general manifolds. Manifolds are treated with positive curvature and negative curvature separately, using extrinsic geometry in the former case and intrinsic geometry in the latter for reasons that will be explained. A unified approach to EL will be developed in each setting. EL is considered for data on Stiefel and Grassmann manifolds, for example. It turns out that EL is straightforward to implement for data from these and other manifolds. Moreover, Wilk's theorem holds in general manifold settings, and bootstrap calibration is available and, under mild conditions, has desirable higher-order properties, as in the Euclidean case.
