Title: The role of finite sample smeariness for directional statistics
Authors: Stephan Huckemann - University of Goettingen (Germany) [presenting]
Benjamin Eltzner - Georg-August-University of Goettingen (Germany)
Shayan Hundrieser - University of Goettingen (Germany)
Abstract: A central tool in nonparametric directional statistics is the Fr'echet mean, and, more generally, so-called generalized Fr'echet means, such as, for instance, principal nested spheres. Even if they are unique - which is a very nontrivial issue that is still to date not satisfactorily solved on spheres of dimension higher than 1 - and even if they exhibit asymptotic normality scaled with the root of sample size, just as their Euclidean kin, this approximation turns out to be invalid in practice in many situations. This phenomenon is called finite sample smeariness (FSS). We give a complete picture of FSS on circles and tori and discuss open questions on spheres. It turns out that the bootstrap can satisfactorily deal with FSS, allowing for statistically controlled inference under FSS. This is not the case in general, however, for classical quantile-based inference. In view of assessing the impact of climate change, we illustrate the role of FSS and inference under the presence of FSS in directional wind data of European cities over the last 20 years.