Conformal accelerations method and efficient evaluation of stable distributions, revisited

TL;DR

This paper presents new efficient methods for evaluating stable distribution functions and quantiles with high accuracy and speed, utilizing integral representations and precomputed characteristic exponent grids.

Contribution

It introduces novel integral representations and computational techniques that significantly improve the efficiency and accuracy of stable distribution evaluations.

Findings

Achieves absolute errors of order 10^{-15} in 0.005-0.1 ms for stable distributions.

Precomputing characteristic exponents enables fast quantile calculations in tails.

Methods outperform popular existing techniques in speed and accuracy.

Abstract

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.