pmvss function

Multivariate Subgaussian Stable Distribution

Computes the probabilities for the multivariate subgaussian stable distribution for arbitrary limits, alpha, shape matrices, and location vectors. See Nolan (2013).

pmvss(
  lower = rep(-Inf, d),
  upper = rep(Inf, d),
  alpha = 1,
  Q = NULL,
  delta = rep(0, d),
  maxpts.pmvnorm = 25000,
  abseps.pmvnorm = 0.001,
  outermost.int = c("stats::integrate", "cubature::adaptIntegrate")[1],
  subdivisions.si = 100L,
  rel.tol.si = .Machine$double.eps^0.25,
  abs.tol.si = rel.tol.si,
  stop.on.error.si = TRUE,
  tol.ai = 1e-05,
  fDim.ai = 1,
  maxEval.ai = 0,
  absError.ai = 0,
  doChecking.ai = FALSE,
  which.stable = c("libstable4u", "stabledist")[1]
)

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.

• maxpts.pmvnorm: Defaults to 25000. Passed to the F_G = pmvnorm() in the integrand of the outermost integral.

• abseps.pmvnorm: Defaults to 1e-3. Passed to the F_G = pmvnorm() in the integrand of the outermost integral.

• outermost.int: select which integration function to use for outermost integral. Default is "stats::integrate" and one can specify the following options with the .si suffix. For diagonal Q, one can also specify "cubature::adaptIntegrate" and use the .ai suffix options below (currently there is a bug for non-diagonal Q).

• subdivisions.si: the maximum number of subintervals. The suffix .si indicates a stats::integrate()

  option for the outermost semi-infinite integral in the product distribution formulation.

• rel.tol.si: relative accuracy requested. The suffix .si indicates a stats::integrate()

  option for the outermost semi-infinite integral in the product distribution formulation.

• abs.tol.si: absolute accuracy requested. The suffix .si indicates a stats::integrate()

  option for the outermost semi-infinite integral in the product distribution formulation.

• stop.on.error.si: logical. If true (the default) an error stops the function. If false some errors will give a result with a warning in the message component. The suffix .si indicates a stats::integrate()

  option for the outermost semi-infinite integral in the product distribution formulation.

• tol.ai: The maximum tolerance, default 1e-5. The suffix .ai indicates a cubature::adaptIntegrate type option for the outermost semi-infinite integral in the product distribution formulation.

• fDim.ai: The dimension of the integrand, default 1, bears no relation to the dimension of the hypercube The suffix .ai indicates a cubature::adaptIntegrate type option for the outermost semi-infinite integral in the product distribution formulation.

• maxEval.ai: The maximum number of function evaluations needed, default 0 implying no limit The suffix .ai indicates a cubature::adaptIntegrate type option for the outermost semi-infinite integral in the product distribution formulation.

• absError.ai: The maximum absolute error tolerated The suffix .ai indicates a cubature::adaptIntegrate type option for the outermost semi-infinite integral in the product distribution formulation.

• doChecking.ai: A flag to be thorough checking inputs to C routines. A FALSE value results in approximately 9 percent speed gain in our experiments. Your mileage will of course vary. Default value is FALSE. The suffix .ai indicates a cubature::adaptIntegrate type option for the outermost semi-infinite integral in the product distribution formulation.

• 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.

Returns

The object returned depends on what is selected for outermost.int. In the case of the default, stats::integrate, the value is a list of class "integrate" with components:

• valuethe final estimate of the integral.
• abs.errorestimate of the modulus of the absolute error.
• subdivisionsthe number of subintervals produced in the subdivision process.
• message"OK" or a character string giving the error message.
• callthe matched call.

Note: The reported abs.error is likely an under-estimate as integrate

assumes the integrand was without error, which is not the case in this application.

Examples

## bivariate
U <- c(1,1)
L <- -U
Q <- matrix(c(10,7.5,7.5,10),2)
mvpd::pmvss(L, U, alpha=1.71, Q=Q)

## trivariate
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)
mvpd::pmvss(L, U, alpha=1.71, Q=Q)

## How `delta` works: same as centering
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)
D <- c(0.75, 0.65, -0.35)
mvpd::pmvss(L-D, U-D, alpha=1.71, Q=Q)
mvpd::pmvss(L  , U  , alpha=1.71, Q=Q, delta=D)

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