Calibrated FFT-based density approximations for αα-stable distributions

An algorithm for the approximation of αα-stable densities is developed and compared with similar approximation methodologies. The proposed approach employs an adaptive Simpson rule for the quadrature of the Fourier inversion integral and asymptotic Bergström series expansions for the tails of the density. It is guaranteed that the approximation integrates precisely to unity which is helpful for numerical maximum-likelihood routines. The accuracy of the algorithm has been verified with respect to the values obtained by Nolan's program STABLE for a grid of parameter values. It is shown that a significant reduction of the computational effort with respect to Nolan's program can be achieved while maintaining a satisfying accuracy.