We demonstrate that for the fast numerical inversion of the (generalized) inverse Gaussian distribution two algorithms based on polynomial interpolation are well-suited. Their precision is close to machine precision and they are much faster than the bisection method recently proposed by Y. Lai. / Series: Research Report Series / Department of Statistics and Mathematics

The acceptance-rejection algorithm is often used to sample from non-standard distributions. For this algorithm to be efficient, however, the user has to create a hat function that majorizes and closely matches the density of the distribution to be sampled from. There are many methods for automatically creating such hat functions, but these methods require that the user transforms the density so that she knows the exact location of the transformed density's inflection points. In this paper, we propose an acceptancerejection algorithm which obviates this need and can thus be used to sample from a larger class of distributions. / Series: Research Report Series / Department of Statistics and Mathematics

Simulation and statistical applications often mimic the behavior of a random phenomenon by way of generating random observations that form a known or empirical probability distribution with estimated parameter values. Generation of such random observations is called Random Variate Generation (RVG). The number of simulation and statistical applications provided on the World Wide Web (Web) is on the rise. To facilitate the development of simulation and statistical applications on the Web by way of reuse, there is a need for providing RVG as a Web service. This research involves the development of such a Web service for RVG, which can be invoked programmatically over the Web by using SOAP over the HyperText Transfer Protocol (HTTP) running on top of the Internet. To provide the RVG Web service, an RVG Web application is developed based on the Java 2 Enterprise Edition (J2EE) architecture. The RVG Web application is engineered by using the IBM WebSphere Studio Application Developer and runs on the IBM WebSphere Application Server. A client simulation and statistical application may call the RVG Web service and request the generation of random variates from 27 probability distributions. In addition, the RVG Web service also provides general statistics, scatter plot, and histogram of the requested random variates. The plots and histograms are created in Scalable Vector Graphics (SVG). The RVG Web service: (a) accepts requests in the Extensible Markup Language (XML) format, which is specified according to a request schema, and (b) sends the results to the client application also in the XML format specified according to a reply schema. The interface specification and access information needed to invoke the RVG Web service are provided in the Web Service Description Language (WSDL) document. Any Web-based simulation or statistical application that needs generation of random variates, their scatter plots and histograms, can invoke the RVG Web service programmatically at http://sunfish.cs.vt.edu/RVGWebService . / Master of Science

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

There arise two problems when the expectation of some function with respect to a nonuniform multivariate distribution has to be computed by (quasi-) Monte Carlo integration: the integrand can have singularities when the domain of the distribution is unbounded and it can be very expensive or even impossible to sample points from a general multivariate distribution. We show that importance sampling is a simple method to overcome both problems. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension. An implementation of these algorithms in C is available from the <a href="http://statmath.wu-wien.ac.at/software/hitro/">authors</a>. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Random variate genration is an important tool in statistical computing. Many programms for simulation or statistical computing (e.g. R) provide a collection of random variate generators for many standard distributions. However, as statistical modeling has become more sophisticated there is demand for larger classes of distributions. Adding generators for newly required distribution seems not to be the solution to this problem. Instead so called automatic (or black-box) methods have been developed in the last decade for sampling from fairly large classes of distributions with a single piece of code. For such algorithms a data about the distributions must be given; typically the density function (or probability mass function), and (maybe) the (approximate) location of the mode. In this contribution we show how such algorithms work and suggest an interface for R as an example of a statistical library. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

The generalized inverse Gaussian distribution has become quite popular in financial engineering. The most popular random variate generator is due to Dagpunar (1989). It is an acceptance-rejection algorithm method based on the Ratio-of-uniforms method. However, it is not uniformly fast as it has a prohibitive large rejection constant when the distribution is close to the gamma distribution. Recently some papers have discussed universal methods that are suitable for this distribution. However, these methods require an expensive setup and are therefore not suitable for the varying parameter case which occurs in, e.g., Gibbs sampling. In this paper we analyze the performance of Dagpunar's algorithm and combine it with a new rejection method which ensures a uniformly fast generator. As its setup is rather short it is in particular suitable for the varying parameter case. (authors' abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Perfect slice sampling is a method to turn Markov Chain Monte Carlo (MCMC) samplers into exact generators for independent random variates. The originally proposed method is rather slow and thus several improvements have been suggested. However, two of them are erroneous. In this article we give a short introduction to perfect slice sampling, point out incorrect methods, and give a new improved version of the original algorithm. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing