Max Planck Institute for Multidisciplinary Science
Trouble in Paradise: Smeary Fréchet Means
The central limit theorem holds trivially for random variables supported on a bounded set of real numbers. One might thus be lead to expect the same to be true for data on compact spaces like the circle, the sphere, or projective spaces. However, it turns out that the opposite is true and many complications arise. In our grand tour of the arising problems, including smeariness, finite sample smeariness, and non-uniqueness of means, we will identify potential obstacles to statistical inference and hypothesis testing. To alleviate these difficulties, we present tools to identify potentially smeary or non-unique means. As an alternative to the Fréchet mean we introduce diffusion means as location statistics, which are intrinsically defined, easily interpretable and often easily computable. We explain why they are less affected by troubling phenomena like smeariness.