Uniformly convergent estimators for Multifractional Hurst functions

Probabilités et Statistique

Lieu: 
Salle séminaire M3-324
Orateur: 
Antoine Ayache
Affiliation: 
Université de Lille
Dates: 
Mercredi, 8 Novembre, 2017 - 10:30 - 11:30
Résumé: 
This a joint work with Julien Hamonien (Université de Lille)

Multifractional processes have been introduced in the 90’s in order to overcome some limitations of the well-known Fractional Brownian Motion (FBM) due to the constancy in time of its Hurst parameter H; in their context, this parameter becomes a Hölder continuous function H(.). Global and local path roughness of a multifractional process are determined by values of H(.); therefore, several authors have been interested in their statistical estimation, starting from discrete variations of the process. Because of the complex dependence structure of variations, showing consistency of estimators is a tricky problem which often requires hard computations.

The main goal of our talk, is to introduce in the setting of the non-anticipative moving average Linear Multifractional Stable Motion (LMSM) with a stability parameter alpha in the interval (1,2], a new strategy for dealing with the latter problem. In contrast with the previous strategies, this new one, does not require to look for sharp estimates of covariances related to variations; roughly speaking, it consists in expressing them in such a way that they become independent up to negligible remainders. Thanks to it, we obtain:

1. for each compact interval I, a strongly consistent estimator of the minimal value of H(.) over I;

2. more importantly, a strongly consistent estimator of the whole function H(.), which converges in the sense of the uniform norm.