mvtnorm: pmvnorm – R documentation

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pmvnorm

Multivariate Normal Distribution

Description

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

Usage

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

Arguments

`lower`	the vector of lower limits of length n.
`upper`	the vector of upper limits of length n.
`mean`	the mean vector of length n.
`corr`	the correlation matrix of dimension n.
`sigma`	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`.
`algorithm`	an object of class `GenzBretz`, `Miwa` or `TVPACK` specifying both the algorithm to be used as well as the associated hyper parameters.
`...`	additional parameters (currently given to `GenzBretz` for backward compatibility issues).

Details

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.

Value

The evaluated distribution function is returned with attributes

`error`	estimated absolute error and
`msg`	status messages.

Source

http://www.sci.wsu.edu/math/faculty/genz/homepage

References

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.

Examples

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)
print(prob)

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)))

mvtnorm

Multivariate Normal and t Distributions

v1.1-1

GPL-2

Authors

Alan Genz [aut], Frank Bretz [aut], Tetsuhisa Miwa [aut], Xuefei Mi [aut], Friedrich Leisch [ctb], Fabian Scheipl [ctb], Bjoern Bornkamp [ctb] (<https://orcid.org/0000-0002-6294-8185>), Martin Maechler [ctb], Torsten Hothorn [aut, cre] (<https://orcid.org/0000-0001-8301-0471>)

Initial release

2020-06-09