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Abstract:
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Frechet mean and variance provide a way of obtaining mean and variance for general metric space valued random variables and can be used for statistical analysis of data objects that lie in abstract spaces where only pairwise distances are available. Examples of such spaces include covariance matrices, graph Laplacians and univariate probability distributions. We derive a central limit theorem for Frechet variance under mild regularity conditions, utilizing empirical process theory, and also provide a consistent estimator of the asymptotic variance. These results lead to a k-sample test for general metric space valued data objects, with emphasis on comparing Frechet means and variances. We examine the finite sample performance of this novel inference procedure through simulation studies for several special cases that include probability distributions and graph Laplacians, which leads to tests to compare populations of networks. The proposed methodology has good finite sample performance in simulations for different kinds of random objects. We illustrate the proposed methods with resting state functional Magnetic Resonance Imaging data.
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