Choice of Algorithm and Hyper Parameters

Choose between three algorithms for evaluating normal distributions and define hyper parameters.

GenzBretz(maxpts = 25000, abseps = 0.001, releps = 0) Miwa(steps = 128) TVPACK(abseps = 1e-6)
maximum number of function values as integer. The internal FORTRAN code always uses a minimum number depending on the dimension. (for example 752 for three-dimensional problems).
absolute error tolerance; for TVPACK only used for dimension 3.
relative error tolerance as double.
number of grid points to be evaluated.

There are three algorithms available for evaluating normal probabilities: The default is the randomized Quasi-Monte-Carlo procedure by Genz (1992, 1993) and Genz and Bretz (2002) applicable to arbitrary covariance structures and dimensions up to 1000.

For smaller dimensions (up to 20) and non-singular covariance matrices, the algorithm by Miwa et al. (2003) can be used as well.

For two- and three-dimensional problems and semi-infinite integration region, TVPACK implements an interface to the methods described by Genz (2004).


An object of class GenzBretz or Miwa defining hyper parameters.


Genz, A. (1992). Numerical computation of multivariate normal probabilities. Journal of Computational and Graphical Statistics, 1, 141--150.

Genz, A. (1993). Comparison of methods for the computation of multivariate normal probabilities. Computing Science and Statistics, 25, 400--405.

Genz, A. and Bretz, F. (2002), Methods for the computation of multivariate t-probabilities. Journal of Computational and Graphical Statistics, 11, 950--971.

Genz, A. (2004), Numerical computation of rectangular bivariate and trivariate normal and t-probabilities, Statistics and Computing, 14, 251--260.

Genz, A. and Bretz, F. (2009), Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics, Vol. 195. Springer-Verlag, Heidelberg.

Miwa, A., Hayter J. and Kuriki, S. (2003). The evaluation of general non-centred orthant probabilities. Journal of the Royal Statistical Society, Ser. B, 65, 223--234.

Mi, X., Miwa, T. and Hothorn, T. (2009). mvtnorm: New numerical algorithm for multivariate normal probabilities. The R Journal 1(1): 37--39.

  • GenzBretz
  • Miwa
Documentation reproduced from package mvtnorm, version 1.0-5, License: GPL-2

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