Convergence rates for estimators of geodesic distances and Frèchet expectations

Consider a sample n={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 o...

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Bibliographic Details
Published in:Journal of applied probability 2018-12, Vol.55 (4), p.1001-1013
Main Authors: Aaron, Catherine, Bodart, Olivier
Format: Article
Language:English
Subjects:
Convergence
Economic models
Independent variables
Mathematics
Normal distribution
Probability
Regularity
Research Papers
Statistics
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Summary:Consider a sample n={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.
ISSN:0021-9002
1475-6072
DOI:10.1017/jpr.2018.66