Abstract
Transformed density rejection is a very flexible method for generating non-uniform random variates. It is based on the acceptance-rejection principle and utilizes a strictly monotone map that transforms the given density into a concave or convex function. Hat function and squeezes are then constructed by means of tangents and secant. We present a new method that works for arbitrary one time continuously differentiable densities. It requires together with the log-density and its derivative a partition of the domain into subdomains that contain at most one inflection point. This improves a previous method of the authors in which also the second derivative is required. We show how the algorithm can be applied to generate from the Generalized Inverse Gaussian distribution, from the Generalized Hyperbolic distribution and from the Watson distribution. The new algorithm can also generate random variates from truncated distributions without problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Barndorff-Nielsen, O.E.: Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. Lon. A 353, 401–419 (1977)
Botev, Z., Belzile, L.: TruncatedNormal: Truncated Multivariate Normal and Student Distributions (2021). https://CRAN.R-project.org/package=TruncatedNormal. R package version 2.2.2
Botev, Z., L’Ecuyer, P.: Simulation from the Normal distribution truncated to an interval in the tail. In: 10th EAI International Conference on Performance Evaluation Methodologies and Tools. ACM (2017). https://doi.org/10.4108/eai.25-10-2016.2266879
Botev, Z.I., L’Ecuyer, P.: Efficient probability estimation and simulation of the truncated multivariate Student-t distribution. In: 2015 Winter Simulation Conference (WSC), pp. 380–391 (2015). https://doi.org/10.1109/WSC.2015.7408180
Botts, C.: A modified adaptive accept-reject algorithm for univariate densities with bounded support. J. Stat. Comput. Simul. 81(3), 1039–1053 (2011)
Botts, C., Hörmann, W., Leydold, J.: Transformed density rejection with inflection points. Stat. Comput. 23(2), 251–260 (2012). https://doi.org/10.1007/s11222-011-9306-4
Dagpunar, J.S.: An easily implemented generalised inverse Gaussian generator. Commun. Stat. - Simul. Comput. 18(2), 703–710 (1989). https://doi.org/10.1080/03610918908812785
Devroye, L.: A simple algorithm for generating random variates with a log-concave density. Computing 33(3–4), 247–257 (1984)
Devroye, L.: Non-Uniform Random Variate Generation. Springer, New-York (1986)
Evans, M., Swartz, T.: Random variable generation using concavity properties of transformed densities. J. Comput. Graph. Stat. 7(4), 514–528 (1998)
Gilks, W.R., Wild, P.: Adaptive rejection sampling for Gibbs sampling. Appl. Stat. 41(2), 337–348 (1992)
Hörmann, W.: A rejection technique for sampling from T-concave distributions. ACM Trans. Math. Softw. 21(2), 182–193 (1995)
Hörmann, W., Leydold, J., Derflinger, G.: Automatic Nonuniform Random Variate Generation. Springer, Berlin (2004)
Leydold, J., Botts, C., Hörmann, W.: Tinflex: A Universal Non-Uniform Random Number Generator (2022). https://CRAN.R-project.org/package=Tinflex. R package version 2.1
Leydold, J., Janka, E., Hörmann, W.: Variants of transformed density rejection and correlation induction. In: Fang, K.T., Hickernell, F.J., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 345–356. Springer, Heidelberg (2002)
Sablica, L., Hornik, K., Leydold, J.: Random sampling from the Watson distribution. Research Report Series/Department of Statistics and Mathematics 134, WU Vienna University of Economics and Business, Vienna (2022). https://epub.wu.ac.at/8582/
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hörmann, W., Leydold, J. (2022). A Generalized Transformed Density Rejection Algorithm. In: Botev, Z., Keller, A., Lemieux, C., Tuffin, B. (eds) Advances in Modeling and Simulation. Springer, Cham. https://doi.org/10.1007/978-3-031-10193-9_14
Download citation
DOI: https://doi.org/10.1007/978-3-031-10193-9_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-10192-2
Online ISBN: 978-3-031-10193-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)