mvtnorm (version 1.1-1)

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).

Value

The evaluated distribution function is returned with attributes

error

estimated absolute error and

msg

status messages.

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.

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.

See Also

qmvnorm

Examples

# NOT RUN {
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)))

# }