Wolfgang Hörmann

Automatic Nonuniform Random Variate Generation in R

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. (authors abstract)

A rejection technique for sampling from T -concave distributions

A rejection algorithm that uses a new method for constructing simple hat functions for a unimodal, bounded density <italic>f</italic> is introduced called “transformed density rejection.” It is based on the idea of transforming <italic>f</italic> with a suitable transformation <italic>T</italic> such that <italic>T(f(x))</italic> is concave. <italic>f</italic> is then called <italic>T</italic>-concave, and tangents of <italic>T(f(x))</italic> in the mode and in a point on the left and right side are used to construct a hat function with a table-mountain shape. It is possible to give conditions for the optimal choice of these points of contact. With <italic>T</italic>= -1/xxx, the method can be used to construct a universal algorithm that is applicable to a large class of unimodal distributions, including the normal, beta, gamma, and <italic>t</italic>-distribution.

Random variate generation by numerical inversion when only the density is known

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.

A sweep-plane algorithm for generating random tuples in simple polytopes

A sweep-plane algorithm by Lawrence for convex polytope computation is adapted to generate random tuples on simple polytopes. In our method an affine hyperplane is swept through the given polytope until a random fraction (sampled from a proper univariate distribution) of the volume of the polytope is covered. Then the intersection of the plane with the polytope is a simple polytope with smaller dimension. In the second part we apply this method to construct a black-box algorithm for log-concave and T-concave multivariate distributions by means of transformed density rejection. (authors abstract)

The transformed rejection method for generating Poisson random variables

The transformed rejection method, a combination of the inversion and the rejection method, which is used to generate non-uniform random numbers from a variety of continuous distributions can be applied to discrete distributions as well. For the Poisson distribution a short and simple algorithm is obtained which is well suited for large values of the Poisson parameter

Automatic random variate generation for simulation input

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A universal generator for discrete log-concave distributions

, even when

A general control variate method for option pricing under Lévy processes

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Generating generalized inverse Gaussian random variates

may vary from call to call. The average number of uniform deviates required is lower than for any of the known uniformly fast algorithms. Timings for a C implementation show that the algorithm needs only half of the code but is - for

Efficient risk simulations for linear asset portfolios in the t-copula model

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