QRM (version 0.4-7)

GHYP: 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.

Value

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

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.

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)