Finite Sample Smeariness of Fréchet Means and Application to Climate
share
Fréchet means on manifolds are minimizers of expected squared distance, just as their Euclidean kin. On manifolds, Fréchet means for most distributions feature a classical asymptotic Gaussian central limit theorem with √(n)-rate, except for some distributions, some of which are known on spheres. They feature slower and non-Gaussian rates, called smeary rates. Here we first describe exhaustively all smeary distributions on circles, comprising not only power smeariness, but also logarithmic smeariness. Then we make the concept of finite sample smeariness (FSS) precise and show for the circle that FSS affects all distributions that are spread on both sides beyond an open half circle. An analog results holds for tori. We statistically quantify the scale of FSS and find two different qualities: If asymptotically, the sample Fréchet variance remains strictly above the population Fréchet variance, we speak of Type I FSS, this is the case, e.g. for all (!) von Mises distributions. Else, we have Type II FSS, e.g. for distributions with support excluding the antipode of its mean. For both types of FSS it turns out that the nominal level of nonparametric asymptotic quantile based tests for the circular mean is higher than the true level. Simulations indicate, however, that suitably designed bootstrap tests preserve the 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.
READ FULL TEXT