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Finite Sample Smeariness on Spheres

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Geometric Science of Information (GSI 2021)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 12829))

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Abstract

Finite Sample Smeariness (FSS) has been recently discovered. It means that the distribution of sample Fréchet means of underlying rather unsuspicious random variables can behave as if it were smeary for quite large regimes of finite sample sizes. In effect classical quantile-based statistical testing procedures do not preserve nominal size, they reject too often under the null hypothesis. Suitably designed bootstrap tests, however, amend for FSS. On the circle it has been known that arbitrarily sized FSS is possible, and that all distributions with a nonvanishing density feature FSS. These results are extended to spheres of arbitrary dimension. In particular all rotationally symmetric distributions, not necessarily supported on the entire sphere feature FSS of Type I. While on the circle there is also FSS of Type II it is conjectured that this is not possible on higher-dimensional spheres.

Acknowledging DFG HU 1575/7, DFG GK 2088, DFG EXC 2067, DFG CRC 803, meteoblue AG, and the Niedersachsen Vorab of the Volkswagen Foundation.

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Correspondence to Benjamin Eltzner .

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Eltzner, B., Hundrieser, S., Huckemann, S. (2021). Finite Sample Smeariness on Spheres. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_2

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  • DOI: https://doi.org/10.1007/978-3-030-80209-7_2

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