In the past decades, the central limit theorem (CLT) has been generalized to non-Euclidean data spaces. Some years ago, it was found that for some random variables on the circle, the sample Fréchet mean fluctuates around the population mean asymptotically at a scale with exponent with a non-normal distribution if the probability density at the antipodal point of the mean is . The author and his collaborator recently discovered that for some random variables on higher dimensional spheres. In this article we show that, even more surprisingly, the phenomenon on spheres of higher dimension is qualitatively different from that on the circle, as it depends purely on geometrical properties of the space, namely its curvature, and not on the density at the antipodal point. This gives rise to the new concept of geometrical smeariness. In consequence, the sphere can be deformed, say, by removing a neighborhood of the antipodal point of the mean and gluing a flat space there, with a smooth transition piece. This yields smeariness on a manifold, which is diffeomorphic to Euclidean space. We give an example family of random variables with 2-smeary mean, that is, with , whose range has a hole containing the cut locus of the mean. The hole size exhibits a curse of dimensionality as it can increase with dimension, converging to the whole hemisphere opposite a local Fréchet mean. We observe smeariness in simulated landmark shapes on Kendall pre-shape space and in real data of geomagnetic north pole positions on the two-dimensional sphere.
The author gratefully acknowledges funding by DFG SFB 755, project B8, DFG SFB 803, project Z2, DFG HU 1575/7 and DFG GK 2088. I am very grateful to Stephan Huckemann for many helpful discussions and detailed comments to the manuscript, Andrew Wood for an inspiring discussion and John Kent and Kanti Mardia for helpful pointers in terms of data application. I would like to thank the anonymous reviewers, whose suggestions helped improve the manuscript.
"Geometrical smeariness – a new phenomenon of Fréchet means." Bernoulli 28 (1) 239 - 254, February 2022. https://doi.org/10.3150/21-BEJ1340