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Convergence rates for estimators of geodesic distances and Frèchet expectations

Part of: Nonparametric inference Multivariate analysis Statistics

Published online by Cambridge University Press:  16 January 2019

Catherine Aaron  and
Olivier Bodart
Catherine Aaron*
Affiliation:
Université Clermont Auvergne
Olivier Bodart*
Affiliation:
Université Jean Monnet
*
* Postal address: Laboratoire de Mathématiques Blaise Pascal (UMR 6620 - CNRS), Université Clermont Auvergne (Clermont-Ferrand 2), 63177 Aubière cedex, France. Email address: [email protected]
** Postal address: Institut Camille Jordan Faculté des Sciences et Techniques, Université Jean Monnet,42023 Saint-Étienne Cedex 2, France.
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Abstract

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Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

Keywords

Geometric inferencegeodesic distancestatistics on manifoldsFréchet expectations

MSC classification

Primary: 62-07: Data analysis
Secondary: 62G05: Estimation 62G20: Asymptotic properties 62H99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1001 - 1013
Copyright
Copyright © Applied Probability Trust 2018 

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