Discrete variations of the fractional Brownian motion in the presence of outliers and an additive noise

Sophie Achard (1) , Jean-François Coeurjolly (2, 1)

1 GIPSA-CICS - GIPSA - Communication Information and Complex Systems
2 SAGAG - Statistique Appliquée et de Géométrie Aléatoire de Grenoble

This paper gives an overview of the problem of estimating the Hurst parameter of a fractional Brownian motion when the data are observed with outliers and/or with an additive noise by using methods based on discrete variations. We show that the classical estimation procedure based on the log-linearity of the variogram of dilated series is made more robust to outliers and/or an additive noise by considering sample quantiles and trimmed means of the squared series or differences of empirical variances. These different procedures are compared and discussed through a large simulation study and are implemented in the \texttt{R} package \texttt{dvfBm}.

Mots clés: fractional Brownian motion; Hurst exponent estimation; discrete variations; robustness; outliers

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]

Sophie Achard, Jean-François Coeurjolly. Discrete variations of the fractional Brownian motion in the presence of outliers and an additive noise. Statistics Surveys, 2010, 4, pp.117-147. ⟨10.1214/09-SS059⟩. ⟨hal-00442760⟩