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

Sebastian Ament, Michael O’Neil

Research output: Research - peer-reviewArticle

### Abstract

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.

Language English (US) 1-15 15 Statistics and Computing 10.1007/s11222-017-9725-y Accepted/In press - Jan 12 2017

### Fingerprint

Stable Distribution
Density Function
Numerical Calculation
Density function
Stable distribution
Probability density function
Integral
Infinitely Divisible Distribution
Cauchy Distribution
Integral form
Characteristic Function
Integral Representation
Numerical Scheme
Asymptotic Expansion
Gaussian distribution
Distribution Function
Probability Distribution

### Keywords

• $$\alpha$$α-Stable
• Infinitely divisible distributions
• Stable distributions

### ASJC Scopus subject areas

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

### Cite this

In: Statistics and Computing, 12.01.2017, p. 1-15.

Research output: Research - peer-reviewArticle

title = "Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics",
abstract = "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.",
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author = "Sebastian Ament and Michael O’Neil",
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doi = "10.1007/s11222-017-9725-y",
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journal = "Statistics and Computing",
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