Confidence intervals for the Hurst parameter of a fractional Brownian motion based on finite sample size

Jean-Christophe Breton (1, 2) , Jean-François Coeurjolly (3, 4)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 MIA - Mathématiques, Image et Applications - EA 3165
3 FIGAL - Fiabilité et Géométrie Aléatoire
4 GIPSA-lab - Grenoble Images Parole Signal Automatique

In this paper, we show how concentration inequalities for Gaussian quadratic form can be used to propose exact confidence intervals of the Hurst index parametrizing a fractional Brownian motion. Both cases where the scaling parameter of the fractional Brownian motion is known or unknown are investigated. These intervals are obtained by observing a single discretized sample path of a fractional Brownian motion and without any assumption on the parameter $H$.

Mots clés: Fractional Brownian motion; Hurst parameter; Concentration inequalities; Confidence intervals

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Jean-Christophe Breton, Jean-François Coeurjolly. Confidence intervals for the Hurst parameter of a fractional Brownian motion based on finite sample size. Statistical Inference for Stochastic Processes, 2012, 15 (1), pp.1-26. ⟨10.1007/s11203-011-9061-3⟩. ⟨hal-00424559v2⟩