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 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 distribution the suggested new algorithms are between 10 and 60 times faster than the algorithms suggested in the literature. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

For discrete distributions a variant of rejection from a continuous hat function is presented. The main advantage of the new method, called rejection-inversion, is that no extra uniform random number to decide between acceptance and rejection is required which means that the expected number of uniform variates required is halved. Using rejection-inversion and a squeeze, a simple universal method for a large class of monotone discrete distributions is developed. It can be used to generate variates from the tails of most standard discrete distributions. Rejection-inversion applied to the Zipf (or zeta) distribution results in algorithms that are short and simple and at least twice as fast as the fastest methods suggested in the literature. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

We present a numerical inversion method for generating random variates from continuous distributions when only the density function is given. The algorithm is based on polynomial interpolation of the inverse CDF and Gauss-Lobatto integration. The user can select the required precision which may be close to machine precision for smooth, bounded densities; the necessary tables have moderate size. Our computational experiments with the classical standard distributions (normal, beta, gamma, t-distributions) and with the noncentral chi-square, hyperbolic, generalized hyperbolic and stable distributions showed that our algorithm always reaches the required precision. The setup time is moderate and the marginal execution time is very fast and nearly the same for all distributions. Thus for the case that large samples with fixed parameters are required the proposed algorithm is the fastest inversion method known. Speed-up factors up to 1000 are obtained when compared to inversion algorithms developed for the specific distributions. This makes our algorithm especially attractive for the simulation of copulas and for quasi-Monte Carlo applications. <P> This paper is the revised final version of the working paper no. 78 of this research report series. / Series: Research Report Series / Department of Statistics and Mathematics

This Online Supplement summarizes our computational experiences with Algorithm NINIGL presented in our paper "Random Variate Generation by Numerical Inversion when only the Density Is Known" (Report No. 90). It is a numerical inversion method for generating random variates from continuous distributions when only the density function is given. The algorithm is based on polynomial interpolation of the inverse CDF and Gauss-Lobatto integration. The user can select the required precision which may be close to machine precision for smooth, bounded densities; the necessary tables have moderate size. Our computational experiments with the classical standard distributions (normal, beta, gamma, t-distributions) and with the noncentral chi-square, hyperbolic, generalized hyperbolic and stable distributions showed that our algorithm always reaches the required precision. The setup time is moderate and the marginal execution time is very fast and nearly the same for all these distributions. Thus for the case that large samples with fixed parameters are required the proposed algorithm is the fastest inversion method known. Speed-up factors up to 1000 are obtained when compared to inversion algorithms developed for the specific distributions. Thus our algorithm is especially attractive for the simulation of copulas and for quasi-Monte Carlo applications. / Series: Research Report Series / Department of Statistics and Mathematics

Applying the ratio-of-uniforms method for generating random variates results in very efficient, fast and easy to implement algorithms. However parameters for every particular type of density must be precalculated analytically. In this paper we show, that the ratio-of-uniforms method is also useful for the design of a black-box algorithm suitable for a large class of distributions, including all with log-concave densities. Using polygonal envelopes and squeezes results in an algorithm that is extremely fast. In opposition to any other ratio-of-uniforms algorithm the expected number of uniform random numbers is less than two. Furthermore we show that this method is in some sense equivalent to transformed density rejection. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing