Alan Genz

Numerical Computation of Multivariate Normal Probabilities

Abstract The numerical computation of a multivariate normal probability is often a difficult problem. This article describes a transformation that simplifies the problem and places it into a form that allows efficient calculation using standard numerical multiple integration algorithms. Test results are presented that compare implementations of two algorithms that use the transformation with currently available software.

Comparison of Methods for the Computation of Multivariate t Probabilities

This article compares methods for the numerical computation of multivariate t probabilities for hyper-rectangular integration regions. Methods based on acceptance-rejection, spherical-radial transformations, and separation-of-variables transformations are considered. Tests using randomly chosen problems show that the most efficient numerical methods use a transformation developed by Genz for multivariate normal probabilities. These methods allow moderately accurate multivariate t probabilities to be quickly computed for problems with as many as 20 variables. Methods for the noncentral multivariate t distribution are also described.

An adaptive algorithm for the approximate calculation of multiple integrals

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 structured to allow ecient implementation on shared memory parallel computers.

Numerical computation of rectangular bivariate and trivariate normal and t probabilities

Algorithms for the computation of bivariate and trivariate normal and t probabilities for rectangles are reviewed. The algorithms use numerical integration to approximate transformed probability distribution integrals. A generalization of Placketts formula is derived for bivariate and trivariate t probabilities. New methods are described for the numerical computation of bivariate and trivariate t probabilities. Test results are provided, along with recommendations for the most efficient algorithms for single and double precision computations.

Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight

Fully symmetric interpolatory integration rules are constructed for multidimensional integrals over infinite 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 efficient and only moderately unstable.

Algorithm 698: DCUHRE: an adaptive multidemensional integration routine for a vector of integrals

J. Berntsen was supported by the Norwegian Research Council for Humanities and Sciences and STATOIL. T. O. Espelid was supported by the Norwegian Research Council for Humanities and Sciences. A Genz was supported by the Norwegian Marshall Fund. Authors’ addresses: J. Berntsen and T. 0. Espelid, Department of Informatics, University of Bergen, Thorm

Numerical computation of multivariate t-probabilities with application to power calculation of multiple contrasts

hlensgate 55, N-5008 Bergen, Norway; A. Genz, Computer Science Department, Washington State University, Pullman, WA 99164-1210. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. @ 1991 ACM 0098-3500/91/1200-0452

On the Numerical Availability of Multiple Comparison Procedures

01.50

A Package for Testing Multiple Integration Subroutines

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)-dimensional hypercube. In this situation standard numerical integration methods can be applied. Three algorithms are discussed in detail. As an application we derive an expression to calculate the power of multiple contrast tests assuming normally distributed data.

Spherical-Radial Integration Rules for Bayesian Computation

In the past many multiple comparison procedure were difficult to perform. Usually, such procedures can be traced back to studentized multiple contrast tests. Numerical difficulties restricted the use of the exact procedures to simple, commonly balanced, designs. Conservative approximations or simulation based approaches have been used in the general cases. However, new efforts and results in the past few years have led to fast and efficient computations of the underlying multidimensional integrals. Inferences for any finite set of linear functions of normal means are now numerically feasible. These include all-pairwise comparisons, comparisons with a control (including dose-response contrasts), multiple comparison with the best, etc. The article applies the numerical progress on multiple comparisons procedures for common balanced and unbalanced designs within the general linear model.