Fast calculation of α-stable density functions based on off-line precomputations. Application to ML parameter estimation

This paper proposes a fast algorithm to compute arbitrary α-stable PDFs and CDFs. The procedure is based on off-line precomputations of α-stable values on a grid of points in the α–β parameter space, as well as in a set of abscissa points. The grid is built by defining two quality measures and assuring that minimum values for these measures are reached on those points. Results indicate that the method here proposed is substantially faster than directly evaluating the standard expressions proposed by Nolan [1]. The proposed algorithm makes it possible to efficiently use estimation methods based on the evaluation of the PDF, such as the Maximum Likelihood (ML) principle, which gives asymptotically unbiased and efficient estimates. To this end, an ML-based estimation algorithm is provided as well.