Uniformly convergent estimators for Multifractional Hurst functions
Probabilités et Statistique
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.
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