Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics

Sebastian Ament, Michael O’Neil

Research output: Contribution to journalArticlepeer-review


Stable distributions are an important class of infinitely divisible probability distributions, of which two special cases are the Cauchy distribution and the normal distribution. Aside from a few special cases, the density function for stable distributions has no known analytic form and is expressible only through the variate’s characteristic function or other integral forms. In this paper, we present numerical schemes for evaluating the density function for stable distributions, its gradient, and distribution function in various parameter regimes of interest, some of which had no preexisting efficient method for their computation. The novel evaluation schemes consist of optimized generalized Gaussian quadrature rules for integral representations of the density function, complemented by asymptotic expansions near various values of the shape and argument parameters. We report several numerical examples illustrating the efficiency of our methods. The resulting code has been made available online.

Original languageEnglish (US)
Pages (from-to)171-185
Number of pages15
JournalStatistics and Computing
Issue number1
StatePublished - Jan 1 2018


  • Generalized Gaussian quadrature
  • Infinitely divisible distributions
  • Numerical quadrature
  • Stable distributions
  • α-Stable

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Computational Theory and Mathematics


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