Josef Leydold
University of Economics and Business Administration
Department for Applied Statistics and Data Processing
Augasse 2-6,
A-1090 Vienna,
Austria
Josef.Leydold@statistik.wu-wien.ac.at
http://statistik.wu-wien.ac.at/staff/leydold/
ACM Transactions on Modeling and Computer Simulation
vol. 8, no. 3 (July 1998)
Paper (PostScript
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Paper (GZipped
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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 -1 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 realsn into simple cones. The hat function is constructed piecewise on each of the cones by tangent hyperplanes. The resulting function is not continuous any more and the rejection constant is bounded from below but the setup and the generation remains quite fast in higher dimensions, e.g. n=8. The paper describes the details how this main idea can be used to construct algorithm TDRMV that generates random tuples from multivariate log-concave distribution with a computable density. Although the developed algorithm is not a real black box method it is adjustable for a large class of log-concave densities.
Algorithms
G.3 [Probability and Statistics]: Random number generation
Rejection method, multivariate log-concave distributions