# pmvnorm

##### Multivariate Normal Distribution

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

- Keywords
- distribution

##### 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

estimated absolute error and

status messages.

##### 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

##### 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)))
# }
```

*Documentation reproduced from package mvtnorm, version 1.1-1, License: GPL-2*