| pmvnorm {mvtnorm} | R Documentation |
Computes the distribution function of the multivariate normal distribution for arbitrary limits and correlation matrices based on algorithms by Genz and Bretz.
pmvnorm(lower=-Inf, upper=Inf, mean=rep(0, length(lower)),
corr=NULL, sigma=NULL, maxpts = 25000, abseps = 0.001,
releps = 0)
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. 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. |
maxpts |
maximum number of function values as integer. |
abseps |
absolute error tolerance as double. |
releps |
relative error tolerance as double. |
This program involves the computation of multivariate normal probabilities with arbitrary correlation matrices. It involves both the computation of singular and nonsingular probabilities. The methodology is described in Genz (1992, 1993).
Note 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.
Multivariate normal density and random numbers are available using
dmvnorm and rmvnorm.
The evaluated distribution function is returned with attributes
error |
estimated absolute error and |
msg |
status messages. |
Fortran Code by Alan Genz <AlanGenz@wsu.edu> and Frank Bretz <frank.bretz@pharma.novartis.com>, R port by Torsten Hothorn <Torsten.Hothorn@rzmail.uni-erlangen.de>
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
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)))