Stochastic modelling of skewed data exhibiting long-range dependence

Time series with long-range dependence appear in many fields including hydrology and there are several studies that have provided evidence of long autocorrelation tails. Provided that the intensity of the long-range dependence in time series of a certain process, quantified by the self-similarity parameter, also known as the Hurst exponent H, could not be falsified, it is then essential that the variable of interest is modelled by a model reproducing long-range dependence. Common models of this category that have been widely used are the fractional Gaussian noise (FGN) and the fractional ARIMA (FARIMA). In case of a variable exhibiting skewness, the previous models can not be implemented in a direct manner. In order to preserve skewness in the simulated series, a normalizing transformation is typically applied in the real-life data at first. The models are then fitted to the normalized data and the produced synthetic series are finally de-normalized. In this paper, a different method is proposed, consisting of two parts. The first one regards the approximation of the long-range dependence by an autoregressive model of high order p AR(p), while the second one regards the direct calculation of the main statistical properties of the random component, that is mean, variance and skewness coefficient. The skewness coefficient calculation of the random component is done using joint sample moments. The advantage of the method is its efficiency and simplicity and the analytical solution.