mvtnorm (version 1.1-2)

pmvnorm: Multivariate Normal Distribution


Computes the distribution function of the multivariate normal distribution for arbitrary limits and correlation matrices.


pmvnorm(lower=-Inf, upper=Inf, mean=rep(0, length(lower)),
        corr=NULL, sigma=NULL, algorithm = GenzBretz(), keepAttr=TRUE, ...)



the vector of lower limits of length n.


the vector of upper limits of length n.


the mean vector of length n.


the correlation matrix of dimension n.


the covariance matrix of dimension n less than 1000. Either corr or sigma can be specified. If sigma is given, the problem is standardized. If neither corr nor sigma is given, the identity matrix is used for sigma.


an object of class GenzBretz, Miwa or TVPACK specifying both the algorithm to be used as well as the associated hyper parameters.


logical indicating if attributes such as error and msg should be attached to the return value. The default, TRUE is back compatible.


additional parameters (currently given to GenzBretz for backward compatibility issues).


The evaluated distribution function is returned, if keepAttr is true, with attributes


estimated absolute error


status message(s).


a character string with class(algorithm).


This program involves the computation of multivariate normal probabilities with arbitrary correlation matrices. It involves both the computation of singular and nonsingular probabilities. The implemented methodology is described in Genz (1992, 1993) (for algorithm GenzBretz), in Miwa et al. (2003) for algorithm Miwa (useful up to dimension 20) and Genz (2004) for the TVPACK algorithm (which covers 2- and 3-dimensional problems for semi-infinite integration regions).

Note the default algorithm GenzBretz is randomized and hence slightly depends on .Random.seed and that both -Inf and +Inf may be specified in lower and upper. For more details see pmvt.

The multivariate normal case is treated as a special case of pmvt with df=0 and univariate problems are passed to pnorm.

The multivariate normal density and random deviates are available using dmvnorm and rmvnorm.


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. (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.

See Also



n <- 5
mean <- rep(0, 5)
lower <- rep(-1, 5)
upper <- rep(3, 5)
corr <- diag(5)
corr[lower.tri(corr)] <- 0.5
corr[upper.tri(corr)] <- 0.5
prob <- pmvnorm(lower, upper, mean, corr)

stopifnot(pmvnorm(lower=-Inf, upper=3, mean=0, sigma=1) == pnorm(3))

a <- pmvnorm(lower=-Inf,upper=c(.3,.5),mean=c(2,4),diag(2))

stopifnot(round(a,16) == round(prod(pnorm(c(.3,.5),c(2,4))),16))

a <- pmvnorm(lower=-Inf,upper=c(.3,.5,1),mean=c(2,4,1),diag(3))

stopifnot(round(a,16) == round(prod(pnorm(c(.3,.5,1),c(2,4,1))),16))

# Example from R News paper (original by Genz, 1992):

m <- 3
sigma <- diag(3)
sigma[2,1] <- 3/5
sigma[3,1] <- 1/3
sigma[3,2] <- 11/15
pmvnorm(lower=rep(-Inf, m), upper=c(1,4,2), mean=rep(0, m), corr=sigma)

# Correlation and Covariance

a <- pmvnorm(lower=-Inf, upper=c(2,2), sigma = diag(2)*2)
b <- pmvnorm(lower=-Inf, upper=c(2,2)/sqrt(2), corr=diag(2))
stopifnot(all.equal(round(a,5) , round(b, 5)))

# }