Бактерівські оцінки параметра Хюрста дробового броунівського руху

Abstract

In the paper consistent estimates of the Hurst parameter of fractional Brownian motion are obtained and confidence intervals of the obtained estimates are constructed. In many applications related to data processing, it is necessary to estimate the Hurst parameter. Among such tasks is the task of signal processing and analysis, when the signal can be considered as the imposition of a useful signal and background noise. Background noise is usually a combination of stochastic and fractal components. Numerical indicators of these properties are, respectively, the Hurst index, the stability index, the coefficients of the relationship of increments, which generalize the autocorrelation function. Obviously, the estimation of the Hurst index is a priority in the analysis of self-similar processes. Currently, there are many methods for estimating the Hurst parameter, but they are all focused on individual cases of processes where the property of self-similarity is combined with either long-term dependence (fractional Brownian motion), or with heavy tails. RS-analysis, disperse-time analysis and deviation analysis are most often used in estimating the Hurst parameter. A common feature of these methods is that they are all based on the use of statistical properties of second-order samples (variance, standard deviation, correlation coefficients).