Statistics Surveys

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

Sophie Achard and Jean-François Coeurjolly

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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 R package dvfBm.

Article information

Statist. Surv., Volume 4 (2010), 117-147.

First available in Project Euclid: 11 June 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 62F10: Point estimation
Secondary: 62F35: Robustness and adaptive procedures

Fractional Brownian motion, Hurst exponent estimation, discrete variations, robustness, outliers


Achard, Sophie; Coeurjolly, Jean-François. Discrete variations of the fractional Brownian motion in the presence of outliers and an additive noise. Statist. Surv. 4 (2010), 117--147. doi:10.1214/09-SS059.

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