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## 2010. “Random variate generation by numerical inversion when only the density is known

Venue: | ACM Transactions on Modeling and Computer Simulation |

Citations: | 10 - 3 self |

### Citations

5183 |
Probability and Measure
- Billingsley
- 1995
(Show Context)
Citation Context ...u) → 0 for n → ∞. That is, the CDFs Fn converge weakly to the CDF F of the target distribution and the corresponding random variates Xn = F−1n (U) converge in distribution to the target distribution [=-=Billingsley 1986-=-]. Random Variate Generation by Numerical Inversion · 5 We are therefore convinced that the u-error is a natural concept for the approximation error of numerical inversion. We use the maximal u-error ... |

1376 | Core Team. 2008. R: A language and environment for statistical computing. in. R Foundation for Statistical Computing - Development |

1017 | Non-Uniform Random Variate Generation - Devroye - 1986 |

366 | A Guide to Simulation - Bratley, Fox, et al. - 1987 |

172 | Hyperbolic distributions in finance,
- Eberlein, Keller
- 1995
(Show Context)
Citation Context ...non-standard distributions. As the setup is fastest for distributions which have a density given by a simple expression we start with trying the hyperbolic distribution which is used in finance, see [=-=Eberlein and Keller 1995-=-]. Here only a quite slow specialized inversion method that reaches maximal u-errors around 10−7 is available in the literature, see [Leobacher and Pillichshammer 2002]. Our automatic algorithm has no... |

149 | Stable distributions - models for heavy tailed data - Nolan - 2010 |

71 | Automatic Nonuniform Random Variate Generation - Hormann, Leydold, et al. - 2004 |

51 | Approximations for Digital Computers - Hastings - 1955 |

41 | Numerical Computing with - Overton - 2001 |

32 |
The general sampling distribution of the multiple correlation coefficient
- Fisher
- 1928
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Citation Context ... also applied our algorithm successfully to several non-standard distributions including the generalized hyperbolic distribution [Barndorff-Nielsen and Blæsild 1983], the noncentral χ2-distribution ([=-=Fisher 1928-=-], see [Johnson et al. 1995]) and the α-stable distribution ([Lévy 1925], see [Nolan 2010] for a recent 20 · G. Derflinger, W. Hörmann, and J. Leydold Table I. Required number of intervals for diffe... |

28 |
The percentile points of distributions having known cumulants.
- Fisher, Cornish
- 1960
(Show Context)
Citation Context ...ution). —Approximate transformations (e.g., Wilson and Hilferty [1931] for χ2 distributions). —general series expansions (e.g., Taylor series or the Cornish-Fisher expansion [Cornish and Fisher 1937; =-=Fisher and Cornish 1960-=-]). —Closed form approximations with polynomials and rational functions. —Continued fractions. —Gaussian and Newton-Cotes quadrature for computing the CDF, and —numerical root finding: Newton’s method... |

27 |
On generating random variates from an empirical distribution
- Chen, Asau
- 1974
(Show Context)
Citation Context ...he distributions) result in an extremely steep inverse CDF and its polynomial interpolation becomes numerically unstable. The sampling part of the algorithm is straightforward. We use indexed search [=-=Chen and Asau 1974-=-] to find the correct interval together with evaluation of the interpolation polynomial. 3.1 Newton’s Interpolation Formula Polynomial interpolation for approximating a function g(x) on some interval ... |

26 | Numerical methods of statistics: - Monahan - 2001 |

24 | Continuous random variate generation by fast numerical inversion - Hörmann, Leydold |

24 | Gaussian random number generators - Thomas, Luk, et al. |

20 | Algorithm as 241: The Percentage Points of the Normal Distribution,” - Wichura - 1988 |

14 |
Computer methods for efficient sampling from largely arbitrary statistical distributions. Computing 26
- Ahrens, H, et al.
- 1981
(Show Context)
Citation Context ...t finding algorithms (e.g., Brent-Dekker method and the bisection method in SSJ [L’Ecuyer 2008]). An alternative approach uses interpolation of tabulated values of the CDF [Hörmann and Leydold 2003; =-=Ahrens and Kohrt 1981-=-]. The tables have to be precomputed in a setup but guarantee fast marginal generation times which are almost independent of the target distribution. Thus such algorithms are well-suited for the fixed... |

14 | Adaptive quadrature—revisited. - GANDER, GAUTSCHI - 2000 |

13 | UNU.RAN { A Library for Non-Uniform Universal Random Variate Generation. Institut fur Statistik - Leydold, Hormann, et al. - 2002 |

10 |
Hyperbolic distributions
- Barndorff-Nielsen, Blæsild
- 1983
(Show Context)
Citation Context ..., beta, gamma, and t-distributions with various parameter settings. We also applied our algorithm successfully to several non-standard distributions including the generalized hyperbolic distribution [=-=Barndorff-Nielsen and Blæsild 1983-=-], the noncentral χ2-distribution ([Fisher 1928], see [Johnson et al. 1995]) and the α-stable distribution ([Lévy 1925], see [Nolan 2010] for a recent 20 · G. Derflinger, W. Hörmann, and J. Leydold ... |

10 | On monotone and convex spline interpolation”, - Costantini - 1986 |

8 | R interface to the UNU.RAN random variate generators, version 0.8. - Leydold, Hörmann - 2008 |

8 |
Simulation accélérée par les méthodes de Monte Carlo et quasi-Monte Carlo: théorie et applications
- Tuffin
- 1997
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Citation Context ...” positions. We consider this deviation as negligible if it is (much) smaller than the resolution of the pseudo-random variate generator. —The same holds for QMC experiments where the F -discrepancy [=-=Tuffin 1997-=-] of a point set {Xi} is computed as discrepancy of the set {F (Xi)}. If the Xi are generated by exact inversion their F -discrepancy coincides with the discrepancy of the underlying low-discrepancy s... |

5 | Numerical methods in scientific computing - Dahlquist, Bjorck - 2008 |

4 |
Moments and cumulants in the specification (f distributions. Rev. de 1 'Inst.. Int. (it tt
- Cornish, Fisher
- 1937
(Show Context)
Citation Context ...he F and the beta distribution). —Approximate transformations (e.g., Wilson and Hilferty [1931] for χ2 distributions). —general series expansions (e.g., Taylor series or the Cornish-Fisher expansion [=-=Cornish and Fisher 1937-=-; Fisher and Cornish 1960]). —Closed form approximations with polynomials and rational functions. —Continued fractions. —Gaussian and Newton-Cotes quadrature for computing the CDF, and —numerical root... |

4 | A method for approximate inversion of the hyperbolic CDF
- Leobacher, Pillichshammer
- 2002
(Show Context)
Citation Context ...stribution which is used in finance, see [Eberlein and Keller 1995]. Here only a quite slow specialized inversion method that reaches maximal u-errors around 10−7 is available in the literature, see [=-=Leobacher and Pillichshammer 2002-=-]. Our automatic algorithm has no problems with that density. Using UNU.RAN the marginal execution time is very fast, almost as fast as the fastest normal generator of UNU.RAN and about 3 times faster... |

3 |
A method for computer generation of variates from arbitrary continuous distributions
- Ulrich, Watson
- 1987
(Show Context)
Citation Context ...etup. (That heuristic worked well for all our experiments but the maximal acceptable error is not guaranteed for all PDFs.) Our algorithm is new, compared to the algorithms of [Ahrens and Kohrt 1981; =-=Ulrich and Watson 1987-=-], as it introduces automatically selected subintervals of variable length as well as a control of the error. Compared to Hörmann and Leydold [2003] the new algorithm has the main practical advantage... |

3 | Algorithm 370. General random number generator - Butler - 1970 |

1 | SSJ: Stochastic Simulation in Java. Départment d’Informatique et de Recherche Opérationnelle (DIRO), Université de Montréal. Version 2.1.1 - L’Ecuyer - 2008 |

1 | Calcul des Probabilités. Gauthier-Villars - Derflinger, Hörmann, et al. - 1925 |

1 | SSJ: Stochastic Simulation in Java. Départment d’Informatique et de Recherche Opérationnelle (DIRO), Université de Montréal. Version 2.1.1 - Derflinger, Hörmann, et al. - 2008 |