An error in the Gaussian random variate generator by Kinderman and Ramage is described that results in the generation of random variates with an incorrect distribution. An additional statement that corrects the original algorithm is given. / Series: Preprint Series / Department of Applied Statistics and Data Processing

The transformed rejection method uses inversion to sample from the dominating density of a rejection algorithm. But in contrast to the usual method it is enough to know the inverse distribution function F^(-1)(x) of the dominating density. This idea can be applied to various continuous (e.g. normal, Cauchy and exponential) and discrete (e.g. binomial and Poisson) distributions with high acceptance probabilities. The resulting algorithms are short, simple and fast. Even more important is the fact that the quality of the method when used in combination with a linear congruential uniform generator is high compared with the quality of the ratio of uniforms method. In addition transformed rejection can be easily employed for correlation induction. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Theoretical considerations and empirical results show that the one-dimensional quality of non-uniform random numbers is bad and the discrepancy is high when they are generated by the ratio of uniforms method combined with linear congruential generators. This observation motivates the suggestion to replace the ratio of uniforms method by transformed rejection (also called exact approximation or almost exact inversion), as the above problem does not occur for this method. Using the function $G(x) =\left( \frac(a)(1-x)+b\right)x $ with appropriate $a$ and $b$ as approximation of the inverse distribution function the transformed rejection method can be used for the same distributions as the ratio of uniforms method. The resulting algorithms for the normal, the exponential and the t-distribution are short and easy to implement. Looking at the number of uniform deviates required, at the code length and at the speed the suggested algorithms are superior to the ratio of uniforms method and compare well with other algorithms suggested in literature. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Very fast automatic rejection algorithms were developed recently which allow to generate random variates from large classes of unimodal distributions. They require the choice of several design points which decompose the domain of the distribution into small sub-intervals. The optimal choice of these points is an important but unsolved problem. So we present an approach that allows to characterize optimal design points in the asymptotic case (when their number tends to infinity) under mild regularity conditions. We describe a short algorithm to calculate these asymptotically optimal points in practice. Numerical experiments indicate that they are very close to optimal even when only six or seven design points are calculated. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

For generating non-uniform random variates, black-box algorithms are powerful tools that allow drawing samples from large classes of distributions. We give an overview of the design principles of such methods and show that they have advantages compared to specialized algorithms even for standard distributions, e.g., the marginal generation times are fast and depend mainly on the chosen method and not on the distribution. Moreover these methods are suitable for specialized tasks like sampling from truncated distributions and variance reduction techniques. We also present a library called UNU.RAN that provides an interface to a portable implementation of such methods. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

This short note discusses performance bounds for "Ahrens" algorithm, that can generate random variates from continuous distributions with monotonically decreasing density. This rejection algorithms uses constant hat-functions and constant squeezes over many small intervals. The choice of these intervals is important. Ahrens has demonstrated that the equal area rule that uses strips of constant area leads to a very simple algorithm. We present bounds on the rejection constant of this algorithm depending only on the number of intervals. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Hierarchical models provide a flexible way of modeling complex behavior. However, the complicated interdependencies among the parameters in the hierarchy make training such models difficult. MCMC methods have been widely used for this purpose, but can often only approximate the necessary distributions. Acceptance-rejection sampling allows for perfect simulation from these often unnormalized distributions by drawing from another distribution over the same support. The efficacy of acceptance-rejection sampling is explored through application to a small dataset which has been widely used for evaluating different methods for inference on hierarchical models. A particular algorithm is developed to draw variates from the posterior distribution. The algorithm is both verified and validated, and then finally applied to the given data, with comparisons to the results of different methods.

The automatic Algorithm ARI developed in this paper can generate variates from a large class of unimodal discrete distributions. It is only necessary to know the mode of the distribution and to have a subprogram available that can evaluate the probabilities. In a set up step the algorithm constructs a table mountain shaped hat function. Then rejection inversion, a new variant of the rejection method for discrete distributions that needs only one uniform random number per iteration, is used to sample from the desired distribution. It is shown that the expeceted number of iterations is uniformly bounded for all T-concave discrete distributions. Utilizing a simple squeeze or an auxiliary table of moderate size, which is initialized during generation and not in the set up, Algorithm ARI is fast, at least as fast as the fastest known methods designed for the Poisson, binomial and hypergeometric distributions. The set up time of the algorithm is not affected by the size of the domain of the distribution and is about ten times longer than the generation of one variate. Compared with the very fast and well known alias and indexed search methods the set up of Algorithm ARI is much faster but the generation time is about two times slower. More important than the speed is the fact that Algorithm ARI is the first automatic algorithm that can generate samples from discrete distributions with heavy tails. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

There exists a vast literature on nonuniform random variate generators. Most of these generators are especially designed for a particular distribution. However in pratice only a few of these are available to practioners. Moreover for problems as (e.g.) sampling from the truncated normal distribution or sampling from fairly uncommon distributions there are often no algorithms available. In the last decade so called universal methods have been developed for these cases. The resulting algorithms are fast and have properties that make them attractive even for standard distributions. In this contribution we describe the concept of Automatic random variate generation where these methods are used to produce a single piece of code in a high level programming language. Using a web-based front-end to such a program this is an easy-to-use source for researchers and programmers for high quality generators for a large class of distributions. Using our UNURAN library we have implemented such a system, which is accessable at <a href="http://statistik.wu-wien.ac.at/anuran" target="_blank">http://statistik.wu-wien.ac.at/anuran</a>. / Series: Preprint Series / Department of Applied Statistics and Data Processing

Generating samples from generalized hyperbolic distributions and non-central chi-square distributions by inversion has become an important task for the simulation of recent models in finance in the framework of (quasi-) Monte Carlo. However, their distribution functions are quite expensive to evaluate and thus numerical methods like root finding algorithms are extremely slow. In this paper we demonstrate how our new method based on Newton interpolation and Gauss-Lobatto quadrature can be utilized for financial applications. Its fast marginal generation times make it competitive, even for situations where the parameters are not always constant. / Series: Research Report Series / Department of Statistics and Mathematics