Abstract. This chapter provides a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.

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.

The inversion method for generating... In this paper we demonstrate that with Hermite interpolation of the inverse CDF we can obtain very small error bounds close to machine precision. Using our adaptive interval splitting method this accuracy is reached with moderately sized tables that allow for a fast and simple generation procedure.

We develop and evaluate algorithms for generating random variates for simulation input. One group called automatic, or black-box algorithms can be used to sample from distributions with known density. They are based on the rejection principle. The hat function is generated automatically in a setup step using the idea of transformed density rejection. There the density is transformed into a concave function and the minimum of several tangents is used to construct the hat function. The resulting algorithms are not too complicated and are quite fast. The principle is also applicable to random vectors. A second group of algorithms is presented that generate random variates directly from a given sample by implicitly estimating the unknown distribution. The best of these algorithms are based on the idea of naive resampling plus added noise. These algorithms can be interpreted as sampling from the kernel density estimates. This method can be also applied to random vectors. There it can be interpreted as a mixture of naive resampling and sampling from the multi-normal distribution that has the same covariance matrix as the data. The algorithms described in this paper have been implemented in ANSI C in a library called UNURAN which is available via anonymous ftp.

this paper we introduce an new approach for universal bounding curves based on the ratio-of-uniforms method. The new algorithm is even simpler and can be applied to a larger class of distributions, including all log-concave distributions. As for Devroye's algorithm the expected number of uniform random numbers does not depend on the particular distribution. In opposition to other black-box algorithms hardly any setup step is required. Thus it is superior in the changing parameter chase.

In this paper we present some variants of transformed density rejection (TDR) that provide more exibility (including the possibility to halve the expected number of uniform random numbers) at the expense of slightly higher memory requirements. Using a synchronized rst stream of uniform variates and a second auxiliary stream (as suggested by Schmeiser and Kachitvichyanukul (1990)) TDR is well suited for correlation induction. Thus high positive and negative correlation between two streams of random variates with same or dierent distributions can be induced.

Different universal (also called automatic or black-box) methods have been suggested to sample from univariate log-concave distributions. The description of a universal generator for bivariate distributions has not been published up to now. The new algorithm for bivariate log-concave distributions is based on the method of transformed density rejection. In order to construct a hat function for a rejection algorithm the bivariate density is transformed by the logarithm into a concave function. Then it is possible to construct a dominating function by taking the minimum of several tangent planes which are by exponentiation transformed back into the original scale. The choice of the points of contact is automated using adaptive rejection sampling. This means that a point that is rejected by the rejection algorithm is used as additional point of contact until the maximal number of points of contact is reached. The paper describes the details how this main idea can be used to construct Al...

This paper presents an overview of the most powerful universal methods. These are based on acceptance/rejection techniques where hat and squeezes are constructed automatically. Although originally motivated to sample from non-standard distributions these methods have advantages that make them attractive even for sampling from standard distributions and thus are an alternative to special generators tailored for particular distributions. Most important are: the marginal generation time is fast and does not depend on the distribution. They can be used for variance reduction techniques, and they produce random numbers of predictable quality. These algorithms are implemented in a library, called UNURAN, which is available by anonymous ftp.

Different universal methods (also called automatic or black-box methods) have been suggested to sample from univariate log-concave distributions. The description of a suitable universal generator for multivariate distributions in arbitrary dimensions has not been published up to now. The new algorithm is based on the method of transformed density rejection. To construct a hat function for the rejection algorithm the multivariate density is transformed by a proper transformation T into a concave function (in the case of log-concave density T (x) = log(x).) Then it is possible to construct a dominating function by taking the minimum of several tangent hyperplanes which are transformed back by T \Gamma1 into the original scale. The domains of different pieces of the hat function are polyhedra in the multivariate case. Although this method can be shown to work, it is too slow and complicated in higher dimensions. In this paper we split the R n into simple cones. The hat function is co...

Abstract. We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions for a particular generator. The algorithms can be implemented in a few lines of high level language code. 1.