Quantiles of the Multivariate Normal Distribution

Computes the equicoordinate quantile function of the multivariate normal distribution for arbitrary correlation matrices based on inversion of pmvnorm, using a stochastic root finding algorithm described in Bornkamp (2018).

Usage

qmvnorm(p, interval = NULL, tail = c("lower.tail", "upper.tail", "both.tails"), 
        mean = 0, corr = NULL, sigma = NULL, algorithm = GenzBretz(),
        ptol = 0.001, maxiter = 500, trace = FALSE, seed = NULL, ...)

Arguments

p: probability.
interval: optional, a vector containing the end-points of the interval to be searched. Does not need to contain the true quantile, just used as starting values by the root-finder. If equal to NULL a guess is used.
tail: specifies which quantiles should be computed. lower.tail gives the quantile \(x\) for which \(P[X \le x] = p\), upper.tail gives \(x\) with \(P[X > x] = p\) and both.tails leads to \(x\) with \(P[-x \le X \le x] = p\).
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 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.
ptol, maxiter, trace: Parameters passed to the stochastic root-finding algorithm. Iteration stops when the 95% confidence interval for the predicted quantile is inside [p-ptol, p+ptol]. maxiter is the maximum number of iterations for the root finding algorithm. trace prints the iterations of the root finder.
seed: an object specifying if and how the random number generator should be initialized, see simulate.
...: additional parameters to be passed to GenzBretz.

Details

Only equicoordinate quantiles are computed, i.e., the quantiles in each dimension coincide. The result is seed dependend. The concept is explained in mvtnorm::Bornkamp:2018.

Value

A list with two components: quantile and f.quantile give the location of the quantile and the difference between the distribution function evaluated at the quantile and p.

References

Examples

qmvnorm(0.95, sigma = diag(2), tail = "both")
#> $quantile
#> [1] 2.236422
#> 
#> $f.quantile
#> [1] -1.310417e-06
#> 
#> attr(,"message")
#> [1] "Normal Completion"