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Register a new userConfidence intervals for the Hurst parameter of a fractional Brownian motion based on finite sample size
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Added by
Jean-Francois Coeurjolly
Publication date
2009
fields
Mathematical Statistics
and research's language is
English
Authors
Jean-Christophe Breton
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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 $H$.
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In this presentation, we introduce a new method for change point analysis on the Hurst index for a piecewise fractional Brownian motion. We first set the model and the statistical problem. The proposed method is a transposition of the FDpV (Filtered Derivative with p-value) method introduced for the detection of change points on the mean in Bertrand et al. (2011) to the case of changes on the Hurst index. The underlying statistics of the FDpV technology is a new statistic estimator for Hurst index, so-called Increment Bernoulli Statistic (IBS). Both FDpV and IBS are methods with linear time and memory complexity, with respect to the size of the series. Thus the resulting method for change point analysis on Hurst index reaches also a linear complexity.
Diffusive transport in many complex systems features a crossover between anomalous diffusion at short times and normal diffusion at long times. This behavior can be mathematically modeled by cutting off (tempering) beyond a mesoscopic correlation time the power-law correlations between the increments of fractional Brownian motion. Here, we investigate such tempered fractional Brownian motion confined to a finite interval by reflecting walls. Specifically, we explore how the tempering of the long-time correlations affects the strong accumulation and depletion of particles near reflecting boundaries recently discovered for untempered fractional Brownian motion. We find that exponential tempering introduces a characteristic size for the accumulation and depletion zones but does not affect the functional form of the probability density close to the wall. In contrast, power-law tempering leads to more complex behavior that differs between the superdiffusive and subdiffusive cases.
In this paper, we will construct the Malliavin derivative and the stochastic integral with respect to the Mixed fractional Brownian motion (mfbm) for H > 1/2. As an application, we try to estimate the drift parameter via Malliavin derivative for surplus process with mixed fractional Brownian motion
In some non-regular statistical estimation problems, the limiting likelihood processes are functionals of fractional Brownian motion (fBm) with Hursts parameter H; 0 < H <=? 1. In this paper we present several analytical and numerical results on the moments of Pitman estimators represented in the form of integral functionals of fBm. We also provide Monte Carlo simulation results for variances of Pitman and asymptotic maximum likelihood estimators.
The recent paper Simple confidence intervals for MCMC without CLTs by J.S. Rosenthal, showed the derivation of a simple MCMC confidence interval using only Chebyshevs inequality, not CLT. That result required certain assumptions about how the estimator bias and variance grow with the number of iterations $n$. In particular, the bias is $o(1/sqrt{n})$. This assumption seemed mild. It is generally believed that the estimator bias will be $O(1/n)$ and hence $o(1/sqrt{n})$. However, questions were raised by researchers about how to verify this assumption. Indeed, we show that this assumption might not always hold. In this paper, we seek to simplify and weaken the assumptions in the previously mentioned paper, to make MCMC confidence intervals without CLTs more widely applicable.
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