Monte Carlo Multivariate Subgaussian Stable Distribution

Computes probabilities of the multivariate subgaussian stable distribution for arbitrary limits, alpha, shape matrices, and location vectors via Monte Carlo (thus the suffix _mc).

Usage

pmvss_mc(
  lower = rep(-Inf, d),
  upper = rep(Inf, d),
  alpha = 1,
  Q = NULL,
  delta = rep(0, d),
  which.stable = c("libstable4u", "stabledist")[1],
  n = NULL
)

Arguments

lower

the vector of lower limits of length n.

upper

the vector of upper limits of length n.

alpha

default to 1 (Cauchy). Must be 0<alpha<2

Q

Shape matrix. See Nolan (2013).

delta

location vector.

which.stable

defaults to "libstable4u", other option is "stabledist". Indicates which package should provide the univariate stable distribution in this production distribution form of a univariate stable and multivariate normal.

n

number of random vectors to be drawn for Monte Carlo calculation.

Value

a number between 0 and 1, the estimated probability via Monte Carlo

References

Nolan JP (2013), Multivariate elliptically contoured stable distributions: theory and estimation. Comput Stat (2013) 28:2067–2089 DOI 10.1007/s00180-013-0396-7

Examples

## print("mvpd (d=2, alpha=1.71):")
U <- c(1,1)
L <- -U
Q <- matrix(c(10,7.5,7.5,10),2)
mvpd::pmvss_mc(L, U, alpha=1.71, Q=Q, n=1e3)
#> [1] 0.051
mvpd::pmvss   (L, U, alpha=1.71, Q=Q)
#> 0.04973221 with absolute error < 4.2e-05

## more accuracy = longer runtime
mvpd::pmvss_mc(L, U, alpha=1.71, Q=Q, n=1e4)
#> [1] 0.0494

U <- c(1,1,1)
L <- -U
Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
## print("mvpd: (d=3, alpha=1.71):")
mvpd::pmvss_mc(L, U, alpha=1.71, Q=Q, n=1e3)
#> [1] 0.017