A simple algorithm for generating random variates with a log-concave density.

*(English)*Zbl 0561.65004A 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

##### MSC:

65C10 | Random number generation in numerical analysis |

62-04 | Software, source code, etc. for problems pertaining to statistics |

65C99 | Probabilistic methods, stochastic differential equations |

##### Keywords:

random variate generation; simulation; log concavity; inequalities; normal density; gamma density; Weibull density; beta density; exponential power density; generalized inverse Gaussian density; random number generator; FORTRAN program
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