Finite sample smeariness of Fréchet means with application to climate

Abstract

Fréchet means on non-Euclidean spaces may exhibit nonstandard asymptotic rates rendering quantile-based asymptotic inference inapplicable. We show here that this affects, among others, all circular distributions whose support is not contained in a closed half circle. We exhaustively describe this phenomenon on the circle and introduce a new concept on metric spaces which we call finite samples smeariness (FSS). In the presence of FSS, for which we provide a test, it turns out that quantile-based tests for equality of Fréchet means systematically feature effective levels higher than their nominal level. This effect can persevere asymptotically for increasing sample size, or may not, depending on the type of FSS. In contrast, suitable bootstrap-based tests correct for FSS and asymptotically attain the correct level. For illustration of relevance we apply our method to directional wind data from two European cities. It turns out that quantile based tests, not correcting for FSS, find a multitude of significant wind changes. This multitude condenses to a few years featuring significant wind changes when our bootstrap tests are applied, correcting for FSS.