Generating a single order statistic without generating the full sample can be an important task for simulations. If the density and the CDF of the distribution are given, then it is no problem to compute the density of the order statistic. In the main theorem it is shown that the concavity properties of that density depend directly on the distribution itself. Especially for log-concave distributions, all order statistics have log-concave distributions themselves. So recently suggested automatic transformed density rejection algorithms can be used to generate single order statistics. This idea leads to very fast generators. For example for the normal and gamma distributions, the suggested new algorithms are between 10 and 60 times faster than the algorithms suggested in the literature.

}, keywords = {black box, order statistics, T-concave, TDR}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @conference {Hoermann;Derflinger:1997a, title = {An automatic generator for a large class of unimodal discrete distributions}, booktitle = {ESM 97}, year = {1997}, pages = {139{\textendash}144}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @article {Hoermann;Derflinger:1996a, title = {Rejection-Inversion to Generate Variates from Monotone Discrete Distributions}, journal = {ACMTOMACS}, volume = {6}, number = {3}, year = {1996}, pages = {169{\textendash}184}, keywords = {monotone discrete distributions, Poisson distribution, random number generation, rejection-inversion, T-concave, universal algorithm, Zipf distribution}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @article {Hoermann;Derflinger:1994a, title = {The transformed rejection method for generating random variables, an alternative to the ratio of uniforms method}, journal = {Commun. Stat., Simulation Comput.}, volume = {23}, number = {3}, year = {1994}, pages = {847-860}, keywords = {normal distribution, random variate generation, t-distribution, transformed rejection method}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @conference {Hoermann;Derflinger:1994b, title = {Universal generators for correlation induction}, booktitle = {Compstat, Proceedings in Computational Statistics}, year = {1994}, pages = {52{\textendash}57}, publisher = {Physica-Verlag}, organization = {Physica-Verlag}, address = {Heidelberg}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @article {Hoermann;Derflinger:1993a, title = {A Portable Random Number Generator Well Suited for the Rejection Method}, volume = {19}, number = {4}, year = {1993}, pages = {489{\textendash}495}, abstract = {Up to now, all known efficient portable implementations of linear congruential random number generators with modulus $2^{31} - 1$ have worked only with multipliers that are small compared with the modulus. We show that for nonuniform distributions, the rejection method may generate random numbers of bad qualify if combined with a linear congruential generator with small multiplier. A method is described that works for any multiplier smaller than $2^{30}$. It uses the decomposition of multiplier and seed in high-order and low-order bits to compute the upper and lower half of the product. The sum of the two halves gives the product of multiplier and seed modulo $2^{21} - 1$. Coded in ANSI-C and FORTRAN77 the method results in a portable implementation of the linear congruential generator that is as fast or faster than other portable methods.

}, keywords = {algorithms, linear congruential generator, portability, quality of nonuniform random numbers, rejection method, uniform random number generator}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} } @article {Hoermann;Derflinger:1990a, title = {The ACR method for generating normal random variables}, journal = {OR Spektrum}, volume = {12}, number = {3}, year = {1990}, pages = {181{\textendash}185}, keywords = {acceptance-complement method, ACR method, decomposition method, normal random number generator, ratio of uniform method}, author = {Wolfgang H{\"o}rmann and Derflinger, Gerhard} }