On approximation theorems for the Euler characteristic with applications to the bootstrap

Krebs, Johannes; Roycraft, Benjamin; Polonik, Wolfgang

doi:10.1214/21-ejs1898

Cited by 7 publications

(1 citation statement)

References 35 publications

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“…Furthermore, Blaszczyszyn et al [2] derived asymptotic normality of geometric statistics (not necessarily U-statistics) when the input process exhibits fast decay of correlations. In the context of random topology, proving the asymptotic normality of the simplex counts, which themselves are U-statistics, will be a crucial step in deriving the central limit theorem for topological invariants, such as the Euler characteristic and Betti numbers [17,21,29,35]. If r n decays more slowly, such that n k r d(k−1) n → c, n → ∞, for some c ∈ (0, ∞), the k-tuples that satisfy the geometric conditions in H n will occur less frequently.…”

Section: Introductionmentioning

confidence: 99%

Limit theory for U-statistics under geometric and topological constraints with rare events

Owada

2022

J. Appl. Probab.

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We study the geometric and topological features of U-statistics of order k when the k-tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate scaling, we establish the convergence of U-statistics in vague topology, while the structure of a non-degenerate limit measure is also revealed. Our general result shows various limit theorems for geometric and topological statistics, including persistent Betti numbers of Čech complexes, the volume of simplices, a functional of the Morse critical points, and values of the min-type distance function. The required vague convergence can be obtained as a result of the limit theorem for point processes induced by U-statistics. The latter convergence particularly occurs in the $\mathcal M_0$ -topology.

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Section: Introductionmentioning

confidence: 99%

Limit theory for U-statistics under geometric and topological constraints with rare events

Owada

2022

J. Appl. Probab.

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Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

Thomas

Owada

2021

Extremes

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Topology-driven goodness-of-fit tests in arbitrary dimensions

Dłotko,

Hellmer,

Stettner

et al. 2023

Stat Comput

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This paper adopts a tool from computational topology, the Euler characteristic curve (ECC) of a sample, to perform one- and two-sample goodness of fit tests. We call our procedure TopoTests. The presented tests work for samples of arbitrary dimension, having comparable power to the state-of-the-art tests in the one-dimensional case. It is demonstrated that the type I error of TopoTests can be controlled and their type II error vanishes exponentially with increasing sample size. Extensive numerical simulations of TopoTests are conducted to demonstrate their power for samples of various sizes.

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On approximation theorems for the Euler characteristic with applications to the bootstrap

Cited by 7 publications

References 35 publications

Limit theory for U-statistics under geometric and topological constraints with rare events

Limit theory for U-statistics under geometric and topological constraints with rare events

Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

Topology-driven goodness-of-fit tests in arbitrary dimensions

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