zbMATH — the first resource for mathematics

A simple algorithm for generating random variates with a log-concave density. (English) Zbl 0561.65004
A large number of densities, as the normal density, the gamma density, the Weibull density, the beta density, the exponential power density, the density of W. F. Perks distribution [On some experiments in the graduation of mortality statistics, J. Inst. Actuaries 58, 12-57 (1932)], the density of J. Talacko’s distribution [Trabajos Estadíst. 7, 159-174 (1956; Zbl 0074.128)], the density of the extreme value distribution, or the generalized inverse Gaussian density [cf. B. Jørgensen, Statistical property of the generalized inverse Gaussian distribution (1980: Zbl 0436.62016)], have the same properties. So they are log-concave, that is to say that their logarithms are concave functions on their support. The importance of generating random variates is well known. The author gives here an algorithm for generating random variates with a log-concave density f on \({\mathbb{R}}\) and known mode in average number of operations independent of f. This algorithm requires the presence of a uniform [0,1] random number generator and a subprogram for computing f. So the author completes his paper with a short FORTRAN program. We remark that uniformly fast algorithms were found for the gamma family first by J. H. Ahrens and U. Dieter [Computing 12, 223-246 (1974; Zbl 0285.65009)] and by P. R. Tadikamalla and M. E. Johnson [Am. J. Math. Manage. Sci. 1, 213-236 (1981; Zbl 0536.65006)], but the algorithm presented here has limited average time and is independent of the density. Well, we can say that this algorithm can be useful for a large number of specialists.
Reviewer: A.Donescu

65C10 Random number generation in numerical analysis
62-04 Software, source code, etc. for problems pertaining to statistics
65C99 Probabilistic methods, stochastic differential equations
Full Text: DOI
[1] Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Tables. New York: Dover Publications 1970. · Zbl 0171.38503
[2] Ahrens, J. H., Dieter, U.: Computer methods for sampling from gamma, beta, Poisson and binomial distributions. Computing12, 223–246 (1974). · Zbl 0285.65009
[3] Best, J. D.: Letter to the editor. Applied Statistics27, 181 (1978).
[4] Cheng, R. C. H.: The generation of gamma variables with non-integral shape parameter. Annals of Statistics26, 71–75 (1977).
[5] Davidovic, Ju. S., Korenbljum, B. I., Hacet, B. I.: A property of logarithmically concave functions. Dokl. Akad. Nauk SSR185, 477–480 (1969). · Zbl 0185.12303
[6] Devroye, L.: On the computer generation of random variates with monotone or unimodal densities. Computing32, 43–68 (1984). · Zbl 0526.65005
[7] Gumbel, E. J.: Statistics of Extremes. New York: Columbia University Press 1958. · Zbl 0086.34401
[8] Ibragimov, I. A.: On the composition of unimodal distributions. Theory of Probability and Its Applications1, 255–260 (1956).
[9] Jorgensen, B.: Statistical Properties of the Generalized Inverse Gaussian Distribution (Lecture Notes in Statistics, Vol. 9). Berlin-Heidelberg-New York: Springer 1982.
[10] Lekkerkerker, C. G.: A property of log-concave functions I, II. Indagationes Mathematicae15, 505–521 (1953). · Zbl 0051.29903
[11] Perks, W. F.: On some experiments in the graduation of mortality statistics. Journal of the Institute of Actuaries58, 12–57 (1932).
[12] Prekopa, A.: On logarithmic concave measures and functions. Acta Scientiarium Mathematicarum Hungarica34, 335–343 (1973). · Zbl 0264.90038
[13] Tadikamalla, P. R., Johnson, M. E.: A complete guide to gamma variate generation. American Journal of Mathematical and Management Sciences1, 213–236 (1981). · Zbl 0536.65006
[14] Talacko, J.: Perks’ distributions and their role in the theory of Wiener’s stochastic variables. Trabajos de Estadistica7, 159–174 (1956). · Zbl 0074.12804
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.