Abstract
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}.
Highlights
Since the pioneer work of Mandelbrot and Ness (1968), the fractional Brownian motion has become widely popular in a theoretical context as well as in a practical one for modelling selfsimilar phenomena
The fractional Brownian motion is an H-self-similar process, that is for all δ > 0, (BH)t∈R =d δH (BH (t))t∈R with autocovariance function behaving like O(|k|2H−2) as |k| → +∞
This paper focuses on fractional Brownian motion (fBm)-type processes by using discrete variations type procedures
Summary
Since the pioneer work of Mandelbrot and Ness (1968), the fractional Brownian motion (fBm) has become widely popular in a theoretical context as well as in a practical one for modelling selfsimilar phenomena. This paper hightlights one class of these methods, namely the method based on discrete variations, which has known great developments these last years This method originates simultaneously from works of Kent and Wood (1997) and Istas and Lang (1997) in the context of locally self-similar Gaussian processes and more deeply in Coeurjolly (2001) in the case of the fractional Brownian motion. This problem has been already considered by Coeurjolly (2008). This paper is accompanied with a simple R package named dvfBm available on the R CRAN (http://cran.r-project.org/)
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