On spheres of dimension at least two, Fréchet means can converge at rate n^{-1/6} even when data avoid the antipodal point, a 'geometrical smeariness' that is absent on the circle.
Finite Sample Smeariness of Fr\'echet Means and Application to Climate
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abstract
Fr\'echet 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 exceeds a half circle. We exhaustively describe this phenomenon and introduce a new concept which we call finite samples smeariness (FSS). In the presence of FSS, it turns out that quantile-based tests for equality of Fr\'echet means systematically feature effective levels higher than their nominal level which perseveres asymptotically in case of Type I FSS. In contrast, suitable bootstrap-based tests correct for FSS and asymptotically attain the correct level. For illustration of the relevance of FSS in real data, 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.
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Geometrical Smeariness -- A new Phenomenon of Fr\'echet Means
On spheres of dimension at least two, Fréchet means can converge at rate n^{-1/6} even when data avoid the antipodal point, a 'geometrical smeariness' that is absent on the circle.