GHYP {QRM}

Uni- and Multivariate Generalized Hyperbolic Distribution

Description

Values of density and random number generation for uni- and multivariate Generalized Hyperbolic distribution in new QRM parameterization (\chi, \psi, \gamma) and in standard parametrization (\alpha, \beta, \delta); univariate only. See pp. 77–81 in QRM. The special case of a multivariate symmetric GHYP is implemented seperately as function dsmghyp().

Usage

dghyp(x, lambda, chi, psi, mu = 0, gamma = 0, log = FALSE) 
dmghyp(x, lambda, chi, psi, mu, Sigma, gamma, log = FALSE)
dsmghyp(x, lambda, chi, psi, mu, Sigma, log = FALSE)
dghypB(x, lambda, delta, alpha, beta = 0, mu = 0, log = FALSE) 
rghyp(n, lambda, chi, psi, mu = 0, gamma = 0)
rmghyp(n, lambda, chi, psi, Sigma, mu, gamma)
rghypB(n, lambda, delta, alpha, beta = 0, mu = 0)

Arguments

alpha

numeric, parameter(s).

beta

numeric, skewness parameter.

chi

numeric, mixing parameter(s).

delta

numeric, parameter(s).

gamma

numeric, skewness parameter(s).

lambda

numeric, mixing parameter(s).

log

logical, should log density be returned; default is FALSE.

mu

numeric, location parameter(s).

n

integer, count of random variates.

psi

numeric, mixing parameter(s).

Sigma

matrix, dispersion matrix for multivaraiate GHYP.

x

vector, values to evaluate density.

Details

The univariate QRM parameterization is defined in terms of parameters \chi, \psi, \gamma instead of the \alpha, \beta, \delta model used by Blaesild (1981). If \gamma = 0, a normal variance mixture where the mixing variable W has a Generalized Inverse Gaussian distribution (GIG) with parameters \lambda, \chi, \psi is given, with heavier tails. If \gamma > 0, a normal mean-variance mixture where the mean is also perturbed to equal \mu + (W * \gamma) which introduces asymmetry as well, is obtained. Values for \lambda and \mu are identical in both QRM and B parameterizations. The dispersion matrix \Sigma does not appear as argument in the univariate case since its value is identically one.

Value

numeric, value(s) of density or log-density (dghyp, dmghyp, dsmghyp and dghypB) or random sample (rghyp, rmghyp, rghypB)

Note

Density values from dgyhp() should be identical to those from dghypB() if the \alpha, \beta, \delta parameters of the B type are translated to the corresponding \gamma, \chi, \psi parameters of the QRM type by formulas on pp 79–80 in QRM.
If \gamma is a vector of zeros, the distribution is elliptical and dsmghyp() is utilised in dmghyp(). If \lambda = (d + 1) / 2, a d-dimensional hyperbolic density results. If \lambda = 1, the univariate marginals are one-dimensional hyperbolics. If \lambda = -1/2, the distribution is Normal Inverse Gaussian (NIG). If \lambda > 0 and \chi = 0, one obtains a Variance Gamma distribution (VG). If one can define a constant \nu such that \lambda = (-1/2) * \nu and \chi = \nu then one obtains a multivariate skewed-t distribution. See p. 80 of QRM for details.

Examples

old.par <- par(no.readonly = TRUE)
par(mfrow = c(2, 2))
ll <- c(-4, 4)
BiDensPlot(func = dmghyp, xpts = ll, ypts = ll, mu = c(0, 0),
           Sigma = equicorr(2, -0.7), lambda = 1, chi = 1, psi = 1,
           gamma = c(0, 0))
BiDensPlot(func = dmghyp, type = "contour", xpts = ll, ypts = ll,
           mu = c(0, 0), Sigma = equicorr(2, -0.7), lambda = 1,
           chi = 1, psi = 1, gamma = c(0, 0))
BiDensPlot(func = dmghyp, xpts = ll, ypts = ll, mu = c(0, 0),
           Sigma = equicorr(2, -0.7), lambda = 1, chi = 1, psi = 1,
           gamma = c(0.5, -0.5))
BiDensPlot(func = dmghyp, type = "contour", xpts = ll, ypts = ll,
           mu = c(0, 0), Sigma = equicorr(2, -0.7), lambda = 1,
           chi = 1, psi = 1, gamma = c(0.5, -0.5))
par(old.par)