Empirical likelihood for manifolds

There has been growing interest in statistical analysis on random objects taking values ina 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 populationobjects on manifolds. We develop the concept of nonparametric likelihood for data on manifoldsand 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 likelihoodstatistic, we present several generalizations of the proposed methodology: two-sampletesting, inference on the Fréchet variance, quasi Bayesian inference, local Fréchet regression,and estimation of the Fréchet mean set. Simulation and real data examples illustrate theusefulness of the proposed methodology and advantage against the conventional Wald test.