Fast Generation of Order Statistics

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.