Fr\'echet Analysis Of Variance For Random Objects

TL;DR

This paper develops a statistical framework using Fréchet mean and variance for analyzing complex data objects in metric spaces, deriving a central limit theorem and proposing tests for comparing multiple populations.

Contribution

It introduces a central limit theorem for Fréchet variance, along with consistent variance estimators, enabling hypothesis testing for diverse metric space valued data.

Findings

The methodology performs well in finite samples across various data types.

Simulation studies validate the effectiveness of the proposed tests.

Applications include analysis of mortality data and brain imaging networks.

Abstract

Fr\'echet 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 devoid of algebraic structure and operations. Examples of such spaces include covariance matrices, graph Laplacians of networks and univariate probability distribution functions. We derive a central limit theorem for Fr\'echet variance under mild regularity conditions, utilizing empirical process theory, and also provide a consistent estimator of the asymptotic variance. These results lead to a test to compare k populations based on Fr\'echet variance for general metric space valued data objects, with emphasis on comparing means and variances. We examine the finite sample performance of this 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 data on mortality profiles of various countries and resting state Functional Magnetic Resonance Imaging data.