# Multivariate Normal Distribution Computes the distribution function of the multivariate normal distribution for arbitrary limits and correlation matrices. ```r pmvnorm(lower=-Inf, upper=Inf, mean=rep(0, length(lower)), corr=NULL, sigma=NULL, algorithm = GenzBretz(), keepAttr=TRUE, seed = NULL, ...) ``` ## 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 internally. If `corr` is given, it is assumed that appropriate standardization was performed by the user. 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. - `keepAttr`: `logical` indicating if `attributes` such as `error` and `msg` should be attached to the return value. The default, `TRUE` is back compatible. - `seed`: an object specifying if and how the random number generator should be initialized, see `simulate`. - ``...``: 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`. `pmvnorm` is based on original implementations by Alan Genz, Frank Bretz, and Tetsuhisa Miwa developed for computing accurate approximations to the normal integral. Users interested in computing log-likelihoods involving such normal probabilities should consider function `lpmvnorm`, which is more flexible and efficient for this task and comes with the ability to evaluate score functions. ## Returns The evaluated distribution function is returned, if `keepAttr` is true, with attributes - **error**: estimated absolute error - **msg**: status message(s). - **algorithm**: a `character` string with `class(algorithm)`. ## 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, T., 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` for quantiles and `lpmvnorm` for log-likelihoods. ## Examples ```r 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))) ```

pmvnorm function