Conformal accelerations method and efficient evaluation of stable distributions, revisited
This work provides a faster and more accurate numerical method for evaluating stable distributions, which are important in finance and signal processing.
The paper introduces new efficient integral representations and methods for evaluating pdfs, cdfs, and quantiles of stable distributions, achieving absolute errors of order 10^{-15} in 0.005-0.1 msec in Matlab. The methods are faster than popular approaches and applicable to other oscillatory integrals.
We introduce new efficient integral representations and methods for evaluation of pdfs, cpds and quantiles of stable distributions. For wide regions in the parameter space, absolute errors of order $10^{-15}$ can be achieved in 0.005-0.1 msec (Matlab implementation), even when the index of the distribution is small or close to 1. For the calculation of quantiles in wide regions in the tails using the Newton or bisection method, it suffices to precompute several hundred values of the characteristic exponent at points of an appropriate grid (conformal principal components) and use these values in formulas for cpdf and pdf, which require a fairly small number of elementary operations. The methods of the paper are applicable to other classes of integrals, especially highly oscillatory ones, and are typically faster than the popular methods.