Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics
| Autor: | Michael O'Neil, Sebastian Ament |
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| Rok vydání: | 2017 |
| Předmět: |
FOS: Computer and information sciences
Statistics and Probability Probability density function 010103 numerical & computational mathematics Statistics - Computation 01 natural sciences Theoretical Computer Science Stable process Normal distribution 010104 statistics & probability symbols.namesake FOS: Mathematics Mathematics - Numerical Analysis 0101 mathematics Computation (stat.CO) Mathematics Probability (math.PR) Mathematical analysis Cauchy distribution Numerical Analysis (math.NA) Gauss–Kronrod quadrature formula Numerical integration Computational Theory and Mathematics symbols Gaussian quadrature Probability distribution Statistics Probability and Uncertainty Mathematics - Probability |
| Zdroj: | Statistics and Computing. 28:171-185 |
| ISSN: | 1573-1375 0960-3174 |
| DOI: | 10.1007/s11222-017-9725-y |
| Popis: | 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 pre-existing 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 various 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. |
| Databáze: | OpenAIRE |
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