[Estimation of the local regularity index of a sample path]
In this Note, we consider processes that satisfied a local Hölder condition with unknown coefficient γ0. We study two families of estimators for γ0: the first one based upon the whole sample path over [0,TN], TN↗∞, and the second one constructed with n observations at sampling rate δn, δn→0 and nδn→∞ as n→∞. For the almost sure convergence, we give the rates for these two families.
Nous considérons dans cette Note, les processus satisfaisant localement une condition de type Hölder avec un coefficient inconnu γ0. Nous étudions deux familles d'estimateurs pour γ0 : l'une basée sur la connaissance globale de la trajectoire sur [0,TN], TN↗∞, et la deuxième fondée sur n observations échantillonnées aux temps iδn avec i=0,…,n−1, δn→0 et nδn→∞ pour n→∞. Nous donnons les vitesses de convergence presque sûre pour ces deux familles distinctes.
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Delphine Blanke 1, 2
@article{CRMATH_2002__334_2_145_0, author = {Delphine Blanke}, title = {Estimation du coefficient de r\'egularit\'e locale d'une trajectoire de processus}, journal = {Comptes Rendus. Math\'ematique}, pages = {145--148}, publisher = {Elsevier}, volume = {334}, number = {2}, year = {2002}, doi = {10.1016/S1631-073X(02)02241-0}, language = {fr}, }
Delphine Blanke. Estimation du coefficient de régularité locale d'une trajectoire de processus. Comptes Rendus. Mathématique, Volume 334 (2002) no. 2, pp. 145-148. doi : 10.1016/S1631-073X(02)02241-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02241-0/
[1] Blanke D., Estimation of local smoothness coefficients for continuous time processes, Statist. Inf. Stochastic Process, 2002, to appear
[2] Nonparametric Statistics for Stochastic Processes. Estimation and Prediction, Lecture Notes in Statist., 110, Springer-Verlag, New York, 1998
[3] Characterizing surface smoothness via estimation of effective fractal dimension, J. Roy. Statist. Soc. Ser. B, Volume 56 (1994) no. 1, pp. 97-113
[4] Inequalities for increments of stochastic processes and moduli of continuity, Ann. Probab., Volume 20 (1992) no. 2, pp. 1031-1052
[5] Quadratic variations and estimation of the local Hölder index of a Gaussian process, Ann. Inst. H. Poincaré Probab. Statist., Volume 33 (1997) no. 4, pp. 407-436
[6] Peltier R.F., Lévy-Véhel J., A new method for estimating the parameter of fractional Brownian motion, Technical Report 2396, I.N.R.I.A., Novembre 1994
[7] Identification of locally self-similar Gaussian process by using convex rearrangements, Publications I.R.M.A., Volume 53 (2000) no. 4 (Université des Sciences et Technologies de Lille)
[8] Estimating the dimension of a fractal, J. Roy. Statist. Soc. Ser. B, Volume 53 (1991), pp. 353-364
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