Automatic Random Variate Generation

Random variate generation is an important tool for stochastic simulation. Fast algorithms for different standard distributions are well known and often used by simulation practitioners. These algorithms are also available in different softwafe libraries. But it is not well known that there are also automatic (sometimes called universal or black-box) algorithms described in the literature. These algorithms are applicable for large families of different distributions, it is only necessary to have some information about the actual distribution, for example the density and the location of the mode. The automatic algorithm uses these informations to compute all necessary constants for generation from that distribution in the so-called set-up step. Then the automatic algorithm is able to generate variates from that distribution. lt is clear that most automatic algorithms are a bit complicated but they have great practical advantages: An algorithm coded and tested only once can do the same as a library of generators for standard distributions. lf random variates from a new distribution are needed it is not necessary to design and code a new generation algorithm. It is enough to check that the new distribution has the properties necessary for the automatic algorithm and to code the density function of the distribution. The rest of the job is done by the automatic algorithm. The main part of the research will be on a recent idea for automatic random variate generation called transformed density rejection. There the density is transformed by a transformation T into a concave function. For a concave function it is easy to construct a dominating function, simply take the pointwise minimum of several tangents. This piecewise linear function transformed back by the inverse transform of T can be used as dominating function (or hat function) for rejection algorithms. Our research in the last years showed that transformed density rejection is a very flexible method that can be used to design quite different automatic generators. One of the main advantages is that it can be even generalized to multidimensional distributions. The main objectives of the research project are the following: For one-dimensional automatic generators the optimal choice of the points of contact will be disussed. The properties of the family of T-concave distributions will be discussed; based on these results new fast algorithms for order-statistics will be designed. For multivariate automatic generators the main emphasis will be laid in the design and coding of fast methods for bi- and multivariate T-concave distributions. lt will also be investigated if it is possible to generalize these methods to non-T-concave distributions as well. Another part of the project is the investigation of the quality of randorn variates generated by the new automatic algorithms. Is the quality of the uniform random number generator used preserved? Are there certain uniform random number generators that cooperate well with the new methods and others that do not produce high quality random variates? What can be done if a multivariate distribution is necessary for a simulation model and some (multivariate) data are available but there is no idea which multivariate distribution could be used? One objective is the design of an algorithm that can generate multivariate randorn variates directly from data. Perhaps the most important part of the project will be the optimal coding of all automatic algorithms. They will be collected in a C-Iibrary together with a detailed documentation. Based on this software library and documentation we plan to write a monograph including all relevant theory on automatic random variate generation methods.