Theory articles

By Agner Fog

Pseudo-Random Number Generators for Vector Processors and Multicore processors

This article describes a pseudo random number generator for large scale Monte Carlo applications, using the vector processing and multiprocessing capabilities of modern computers. Discusses theoretical problems as well as practical implementation.
Published in Journal of Modern Applied Statistical Methods 14, no. 1 (2015): 308–34.
File name: randomvector.pdf, size: 715745, last modified: 2015-Dec-27.
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Chaotic random number generators with random cycle lengths

This article is a theoretical analysis of fast random number generators based on the chaotic behavior of random maps, such as the RANROT generator.
The article includes an analysis of the distribution of cycle lengths, experimental and theoretical analysis of the randomness, including a spectral test, implementation of a self-test facility, and a discussion of optimizing random number generators for speed.
File name: chaosran.pdf, size: 111804, last modified: 2002-Feb-13.
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Calculation Methods for Wallenius' Noncentral Hypergeometric Distribution

Taking colored balls from an urn without replacement, with bias, gives a probability distribution called Wallenius' noncentral hypergeometric distribution. Several different methods for calculating probabilities from Wallenius' noncentral hypergeometric distribution are derived. Range of applicability, numerical problems and efficiency are discussed for each method. Approximations to the mean and variance are also discussed. This distribution has important applications in models of biased sampling and in models of evolutionary systems. Comparisons are made with a similar distribution called Fisher's Noncentral Hypergeometric Distribution. These two distributions are often confused in the literature and given the same name. The nomenclature problems are discussed.
A revised version of this article is published in Communications In statictics, Simulation and Computation, 2008, vol. 37, no. 2, pp. 258-273.
File name: nchyp1.pdf, size: 197910, last modified: 2008-Feb-06.
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Sampling Methods for Wallenius' and Fisher's Noncentral Hypergeometric distributions

Several methods for generating random variables with univariate and multivariate Wallenius' and Fisher's noncentral hypergeometric distributions are developed. Methods for the univariate distributions include: simulation of urn experiments, inversion by binary search, inversion by chop-down search from the mode, ratio-of-uniforms rejection method, and rejection by sampling in the t domain. Methods for the multivariate distributions include: simulation of urn experiments, conditional method, Gibbs sampling, and Metropolis-Hastings sampling. These methods are useful for Monte Carlo simulation of models of biased sampling and models of evolution and for calculating moments and quantiles of the distributions.
A revised version of this article is published in Communications In statictics, Simulation and Computation, 2008, vol. 37, no. 2, pp. 241-257.
File name: nchyp2.pdf, size: 193043, last modified: 2008-Feb-06.
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