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} } @article {Leydold;Hoermann:1998a, title = {A Sweep-Plane Algorithm for Generating random tuples in simple polytopes}, journal = {Mathematics of Computation}, volume = {67}, number = {224}, year = {1998}, pages = {1617{\textendash}1635}, keywords = {black box, multivariate log-concave distributions, polytope, rejection method, sweep-plane algorithm, T-concave, TDR, uniform distributions, universal method}, author = {Leydold, Josef and Wolfgang H{\"o}rmann} } @article {Hoermann:1995a, title = {A Rejection Technique for Sampling from {T}-Concave Distributions}, journal = {ACMTOMS}, volume = {21}, number = {2}, year = {1995}, pages = {182{\textendash}193}, keywords = {black box, log-concave, nonuniform RNG, rejection algorithm, rejection method, T-concave, t-distribution, transformed density rejection, unimodal distributions}, author = {Wolfgang H{\"o}rmann} }