### 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) |
---|---|

Pages | 1-15 |

Number of pages | 15 |

Journal | Statistics and Computing |

DOIs | |

State | Accepted/In press - Jan 12 2017 |

### Fingerprint

### Keywords

- $$\alpha $$α-Stable
- Generalized Gaussian quadrature
- Infinitely divisible distributions
- Numerical quadrature
- Stable distributions

### ASJC Scopus subject areas

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

### Cite this

**Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics.** / Ament, Sebastian; O’Neil, Michael.

Research output: Research - peer-review › Article

}

TY - JOUR

T1 - Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics

AU - Ament,Sebastian

AU - O’Neil,Michael

PY - 2017/1/12

Y1 - 2017/1/12

N2 - 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.

AB - 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.

KW - $$\alpha $$α-Stable

KW - Generalized Gaussian quadrature

KW - Infinitely divisible distributions

KW - Numerical quadrature

KW - Stable distributions

UR - http://www.scopus.com/inward/record.url?scp=85009236642&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009236642&partnerID=8YFLogxK

U2 - 10.1007/s11222-017-9725-y

DO - 10.1007/s11222-017-9725-y

M3 - Article

SP - 1

EP - 15

JO - Statistics and Computing

T2 - Statistics and Computing

JF - Statistics and Computing

SN - 0960-3174

ER -