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The numerical computation of a multivariate normal probability is often a diicult problem. This article describes a transformation that simpliies the problem and places it into a form that allows eecient calculation using standard numerical multiple integration algorithms. Test results are presented that compare implementations of two algorithms that use… (More)
An adaptive algorithm for numerical integration over hyperrectangular regions is described. The algorithm uses a globally adaptive subdivision strategy. Several precautions are introduced in the error estimation in order to improve the reliability. In each dimension more than one integration rule is made available to the user. The algorithm has been… (More)
Stochastic integration rules are derived for infinite integration intervals, generalizing rules developed by Siegel and O'Brien [SIAM J. Sci. Statist. Comput., 6 (1985), pp. 169–181] for finite intervals. Then random orthogonal transformations of rules for integrals over the surface of the unit m-sphere are used to produce stochastic rules for these… (More)
A new method to calculate the multivariate t-distribution is introduced. We provide a series of substitutions, which transform the starting q-variate integral into one over the q-1 1 6 dimensional hypercube. In this situation standard numerical integration methods can be applied. Three algorithms are discussed in detail. As an application we derive an… (More)
Fully symmetric interpolatory integration rules are constructed for multidimensional inte-grals over innnite integration regions with a Gaussian weight function. The points for these rules are determined by successive extensions of the one dimensional three point Gauss-Hermite rule. The new rules are shown to be eecient and only moderately unstable.
This paper compares acceptance-rejection sampling and methods of Dee ak, Genz and Schervish for the numerical computation of multivariate normal probabilities. Tests using randomly chosen problems show that the most ee-cient numerical methods use a transformation developed by Genz (1992). The methods allow moderately accurate multivariate normal… (More)
Maintainer Torsten Hothorn <Torsten.Hothorn@R-project.org> Description Computes multivariate normal and t probabilities,quantiles, random deviates and densities.
Many statistical multiple integration problems involve integrands that have a dominant peak. In applying numerical methods to solve these problems, statisticians have paid relatively little attention to existing quadrature methods and available software developed in the numerical analysis literature. One reason these methods have been largely overlooked ,… (More)