pmvnorm {mvtnorm} R Documentation

## Multivariate Normal Distribution

### Description

Computes the distribution function of the multivariate normal distribution for arbitrary limits and correlation matrices based on algorithms by Genz and Bretz.

### Usage

```pmvnorm(lower=-Inf, upper=Inf, mean=rep(0, length(lower)),
corr=NULL, sigma=NULL, maxpts = 25000, abseps = 0.001,
releps = 0)
```

### 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. 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.

### 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 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`.

### Value

The evaluated distribution function is returned with attributes

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

### Author(s)

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>

### 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

`qmvnorm`

### 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)))

```

[Package mvtnorm version 0.7-2 Index]