Empirical likelihood for manifolds

There has been growing interest in statistical analysis of random objects taking values in a non-Euclidean metric space. One important class of such objects consists of data on manifolds. This article is concerned with inference on the Fréchet mean and related population objects on manifolds. We develop the concept of nonparametric likelihood for data on manifolds and propose general inference methods by adapting the theory of empirical likelihood. In addition to the basic asymptotic properties, such as Wilks’ theorem of the empirical likelihood statistic, we present several generalizations of the proposed methodology: two-sample testing, inference on the Fréchet variance, quasi-Bayesian inference, local Fréchet regression, and estimation of the Fréchet mean set. Simulation and real data examples illustrate the usefulness of the proposed methodology and its advantage against the conventional Wald test.