pmvss: Multivariate Subgaussian Stable Distribution

In mvpd: Multivariate Product Distributions for Elliptically Contoured Distributions

Multivariate Subgaussian Stable Distribution

Description

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

Usage

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.

Value

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:

• value the final estimate of the integral.

• abs.error estimate of the modulus of the absolute error.

• subdivisions the number of subintervals produced in the subdivision process.

• message "OK" or a character string giving the error message.

• call the 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.

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

## 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)

mvpd documentation built on Sept. 3, 2023, 5:07 p.m.