Stable distributions have received great interest over the last few years in the signal processing community and have proved to be strong alternatives to the Gaussian distribution. There have been several works in the literature addressing the problem of estimating the parameters of stable distributions. However, most 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. Previous work on estimating the parameters of general (possibly skewed) stable distributions has been limited. The existing techniques are either computationally too expensive or their estimates have high variances.
In this paper, we solve the general problem of stable parameter estimation analytically. We introduce three novel classes of estimators for the parameters of general stable distributions. These new classes of estimators are based on formulas we have developed for the fractional and negative order moments of skewed stable random variables. These are generalisations of methods previously suggested for paramater estimation with symmetric stable distributions.
Of all known techniques for the general problem, only the characteristic function technique and the methods we have suggested yield closed form estimates for the parameters which may be efficiently computed. Simulation results show that at least one of our new estimators has better performance than the characteristic function technique over most of the parameter space. Furthermore our techniques require substantially less computation.