Fractional Brownian Bridge as a Tool for Short Time Series Analysis

Traditional fractional stochastic processes represent suitable models for fractal analysis of long time series. However, due to their asymptotic behaviour, the estimation of Hurst exponent is often biased when the sample is too short. The novel approach is based on the construction of fractional Brownian bridge and thanks to its statistical properties and artificial extension to infinite length, it can be used for short time series investigation and resulting estimate was proven not to be burdened by bias. At first, the input signal is split into short stationary segments and the optimal interval length can be obtained via multiple statistical testing. Subsequently, the estimation of the Hurst exponent and its standard deviation is performed on the interval level. The methodology is applied to the stock market indices and based on the Hurst exponent variability in time, the decision about its predictability can be made. As a referential technique, the revisited zero-crossing method is presented and its performance is discussed in the context of obtained results.