Fréchet Covariance and MANOVA Tests for Random Objects in Multiple Metric Spaces

In this manuscript we consider random objects being measured in multiple metric spaces, which may arise when those objects may be measured in multiple distinct ways. In this new multivariate setting, we define a Fr\échet covariance and Fr\échet correlation in two metric spaces, and a Fr\échet covariance matrix and Fr\échet correlation matrix in an arbitrary number of metric spaces. We prove consistency for the sample Fr\échet covariance, and propose several tests to compare the means and covariance matrices between two or more groups. Lastly, we investigate the power and Type I error of each test under a variety of scenarios.