Intrinsic means on the circle: uniqueness, locus and asymptotics
By Thomas Hotz, Stephan F. Huckemann. 2011
79 citations recorded by the source corpus.
This paper gives a comprehensive treatment of local uniqueness, asymptotics and numerics for intrinsic sample means on the circle. It turns out that local uniqueness as well as rates of convergence are governed by the distribution near the antipode. If the distribution is locally less than uniform there, we have local uniqueness and asymptotic normality with a square-root rate. With increased proximity to the uniform distribution the rate can be arbitrarily slow, and in the limit, local uniqueness is lost. Further, we give general distributional conditions, e.g., unimodality, that ensure global uniqueness. Along the way, we discover that sample means can occur only at the vertices of a regular polygon which allows to compute intrinsic sample means in linear time from sorted data. This algorithm is finally applied in a simulation study demonstrating the dependence of the convergence rates on the behavior of the density at the antipode.
Subfields: Artificial Intelligence, Geometry and Topology, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/f08d1eb4652f36662600215d715a06f63c1b73df
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In Artificial Intelligence
In Geometry and Topology
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Citations in the corpus
- A smeary central limit theorem for manifolds with application to high-dimensional spheres (2018, background): “(2012); Hotz and Huckemann (2015), as long as 2πf(−π) < 1, the BP-CLT (3) remains valid. Further, in Hotz and Huckemann (2015) it was shown that 1 = 2πf(−π) is possible, so that the BP-CLT (3), which, under square integrability, holds universally for Euclidean spaces, is wrong for such 2D vectors co”