Discussion of 'Statistics on Manifolds...' by R. Bhattacharya and V. Patrangenaru

摘要

Data, or statistics computed from data, may take values in restricted algebraic structures that form singular subsets of a larger Euclidean space. Older examples include directional or axial data and the space of covariance matrices. Newer examples treated in this paper include various types of similarity shapes and projective shapes that arise in applications, including analysis of digital images. Statistical methodologies for data in the embedding Euclidean space are occasionally effective on manifold-constrained data. For example, the affinely invariant confidence sets for an unknown multivariate distribution that are constructed in Beran and Millar (1986) remain natural when restricted to distributions supported on a sphere. More often, classical techniques of multivariate analysis lack pertinence to samples on manifolds. Professors Bhattacharya and Patrangenaru have provided a welcome survey of hard-earned progress toward effective statistical analysis of manifold-valued data using Fréchet means.