Short universal generators via generalized ratio-of-uniforms method

Leydold, Josef

doi:10.1090/S0025-5718-03-01511-4

Short universal generators via generalized ratio-of-uniforms method
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by Josef Leydold PDF

Math. Comp. 72 (2003), 1453-1471 Request permission

Abstract:

We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions for a particular generator. The algorithms can be implemented in a few lines of high level language code.

References

Ahrens, J. H. (1993). Sampling from general distributions by suboptimal division of domains. Grazer Math. Berichte 319, 20 pp.
J. H. Ahrens, A one-table method for sampling from continuous and discrete distributions, Computing 54 (1995), no. 2, 127–146 (English, with English and German summaries). MR 1318330, DOI 10.1007/BF02238128
Derflinger, G. (2000). private communication.
Derflinger, G., W. Hörmann, and J. Leydold (2002). Universal methods of nonuniform random variate generation. in preparation.
Luc Devroye, On the use of probability inequalities in random variate generation, J. Statist. Comput. Simulation 20 (1984), no. 2, 91–100. MR 775472, DOI 10.1080/00949658408810759
L. Devroye, A simple algorithm for generating random variates with a log-concave density, Computing 33 (1984), no. 3-4, 247–257 (English, with German summary). MR 773927, DOI 10.1007/BF02242271
Luc Devroye, Nonuniform random variate generation, Springer-Verlag, New York, 1986. MR 836973, DOI 10.1007/978-1-4613-8643-8
L. Devroye, A simple generator for discrete log-concave distributions, Computing 39 (1987), no. 1, 87–91 (English, with German summary). MR 908559, DOI 10.1007/BF02307716
Dieter, U. (1989). Mathematical aspects of various methods for sampling from classical distributions. In E. A. Mc Nair, K. J. Musselman, and P. Heidelberger (Eds.), Proc. 1989 Winter Simulation Conf., pp. 477–483.
Evans, M. and T. Swartz (1998). Random variable generation using concavity properties of transformed densities. Journal of Computational and Graphical Statistics 7(4), 514–528.
Gilks, W. R. and P. Wild (1992). Adaptive rejection sampling for Gibbs sampling. Applied Statistics 41(2), 337–348.
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 5th ed., Academic Press, Inc., Boston, MA, 1994. Translation edited and with a preface by Alan Jeffrey. MR 1243179
Wolfgang Hörmann, A rejection technique for sampling from $T$-concave distributions, ACM Trans. Math. Software 21 (1995), no. 2, 182–193. MR 1342355, DOI 10.1145/203082.203089
Hörmann, W. and G. Derflinger (1996). Rejection-inversion to generate variates from monotone discrete distributions. ACM TOMACS 6(3), 169–184.
Hörmann, W. and G. Derflinger (1997). An automatic generator for a large class of unimodal discrete distributions. In A. R. Kaylan and A. Lehmann (Eds.), ESM 97, pp. 139–144.
Kinderman, A. J. and F. J. Monahan (1977). Computer generation of random variables using the ratio of uniform deviates. ACM Trans. Math. Software 3(3), 257–260.
Leydold, J. (2000a). Automatic sampling with the ratio-of-uniforms method. ACM Trans. Math. Software 26(1), 78–98.
J. Leydold, A note on transformed density rejection, Computing 65 (2000), no. 2, 187–192. MR 1807716, DOI 10.1007/s006070070019
Leydold, J. (2001). A simple universal generator for continuous and discrete univariate T-concave distributions. ACM Trans. Math. Software 27(1), 66–82.
Ernst Stadlober, Sampling from Poisson, binomial and hypergeometric distributions: ratio of uniforms as a simple and fast alternative, Berichte der Mathematisch-Statistischen Sektion in der Forschungsgesellschaft Joanneum [Reports of the Mathematical-Statistical Section of the Research Society Joanneum], vol. 303, Forschungszentrum Graz, Mathematisch-Statistische Sektion, Graz, 1989. MR 996245
Wakefield, J. C., A. E. Gelfand, and A. F. M. Smith (1991). Efficient generation of random variates via the ratio-of-uniforms method. Statist. Comput. 1(2), 129–133.