Distribution function of the multivariate Student distribution for arbitrary limits

This function computes the distribution function of a multivariate normal distribution vector for an arbitrary rectangular region [lb, ub]. pmvt computes an estimate and the value is returned along with a relative error and a deterministic upper bound of the distribution function of the multivariate normal distribution. Infinite values for vectors \(u\) and \(l\) are accepted. The Monte Carlo method uses sample size \(n\): the larger the sample size, the smaller the relative error of the estimator.

pmvt(
  mu,
  sigma,
  df,
  lb = -Inf,
  ub = Inf,
  type = c("mc", "qmc"),
  log = FALSE,
  B = 10000
)

Arguments

mu	vector of location parameters
sigma	scale matrix
df	degrees of freedom
lb	vector of lower truncation limits
ub	vector of upper truncation limits
type	string, either of mc or qmc for Monte Carlo and quasi Monte Carlo, respectively
log	logical; if TRUE, probabilities and density are given on the log scale.
B	number of replications for the (quasi)-Monte Carlo scheme

References

Z. I. Botev and P. L'Ecuyer (2015), Efficient probability estimation and simulation of the truncated multivariate Student-t distribution, Proceedings of the 2015 Winter Simulation Conference, pp. 380-391

Examples

d <- 15; nu <- 30;
l <- rep(2, d); u <- rep(Inf, d);
sigma <- 0.5 * matrix(1, d, d) + 0.5 * diag(1, d);
est <- pmvt(lb = l, ub = u, sigma = sigma, df = nu)
# mvtnorm::pmvt(lower = l, upper = u, df = nu, sigma = sigma)
if (FALSE) {
d <- 5
sigma <- solve(0.5 * diag(d) + matrix(0.5, d, d))
# mvtnorm::pmvt(lower = rep(-1,d), upper = rep(Inf, d), df = 10, sigma = sigma)[1]
pmvt(lb = rep(-1, d), ub = rep(Inf, d), sigma = sigma, df = 10)
}