Convergence rates for estimators of geodesic distances and Fréchet expectations
Résumé
Consider a sample Xn = {X1 ,. .. , X n } of i.i.d variables drawn with a probability distribution P supported on a set M ⊂ R d. This article mainly deals with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M , a general convergence result is proved. Assuming M to be a man-ifold of known dimension d ≤ d, and under regularity assumptions on P X , an explicit convergence rate is given. In the case when M has no boundary, the knwoldege of the dimension d is unnecessary 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.
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