dmvss {mvpd}

Multivariate Subgaussian Stable Density

Description

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

Usage

dmvss(
  x,
  alpha = 1,
  Q = NULL,
  delta = rep(0, d),
  outermost.int = c("stats::integrate", "cubature::adaptIntegrate")[1],
  spherical = FALSE,
  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

x

vector of quantiles.

alpha

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

Q

Shape matrix. See Nolan (2013).

delta

location vector

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-diagnoal Q).

spherical

default is FALSE. If true, use the spherical transformation. Results will be identical to spherical = FALSE but may be faster.

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, John P. "Multivariate elliptically contoured stable distributions: theory and estimation." Computational Statistics 28.5 (2013): 2067-2089.

Examples

## print("mvsubgaussPD (d=2, alpha=1.71):")
Q <- matrix(c(10,7.5,7.5,10),2)
mvpd::dmvss(x=c(0,1), alpha=1.71, Q=Q)

## more accuracy = longer runtime
mvpd::dmvss(x=c(0,1),alpha=1.71, Q=Q, abs.tol=1e-8)

Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
## print("mvsubgausPD (d=3, alpha=1.71):")
mvpd::dmvss(x=c(0,1,2), alpha=1.71, Q=Q)
mvpd::dmvss(x=c(0,1,2), alpha=1.71, Q=Q, spherical=TRUE)

## How `delta` works: same as centering
X <- c(1,1,1)
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::dmvss(X-D, alpha=1.71, Q=Q)
mvpd::dmvss(X  , alpha=1.71, Q=Q, delta=D)