| 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, algorithm = GenzBretz(), ...)
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. |
algorithm |
an object of class GenzBretz or Miwa
specifying both the algorithm to be used as well as
the associated hyper parameters. |
... |
additional parameters (currently given to GenzBretz for
backward compatibility issues). |
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. Randomized quasi-Monte Carlo methods are used for the
computations. 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.
The evaluated distribution function is returned with attributes
error |
estimated absolute error and |
msg |
status messages. |
http://www.sci.wsu.edu/math/faculty/genz/homepage
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. (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.
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)))