Density parameter estimation of skewed alfa-stable distributions

Over the last few years, there has been a great interest in alpha-stable distributions for modelling impulsive data. As a critical step in modelling with alpha-stable distributions, the problem of estimating the parameters of stable distributions have been addressed by several works in the literature. However, many of these works consider only the special case of symmetric stable random variables. This is an important restriction though, since most real life signals are skewed. The existing techniques on estimating skewed distribution parameters are either computationally too expensive, require lookup tables or have poor convergence properties. In this paper, we introduce three novel classes of estimators for the parameters of general stable distributions, which are generalisations of methods previously suggested for parameter estimation with symmetric stable distributions. These estimators exploit expressions we develop for fractional lower order, negative order and logarithmic moments and tail statistics. We also introduce simple transformations which allow one to use existing symmetric stable parameter estimation techniques. Techniques suggested in this paper provide the only closed form solutions we are aware of for parameters which may be efficiently computed. Simulation results show that at least one of our new estimators has better performance than the existing techniques over most of the parameter space. Furthermore our techniques require substantially less computation.