ghyp-constructors: Create generalized hyperbolic distribution objects

In ghyp: Generalized Hyperbolic Distribution and Its Special Cases

Create generalized hyperbolic distribution objects

Description

Constructor functions for univariate and multivariate generalized hyperbolic distribution objects and their special cases in one of the parametrizations “chi/psi”, “alpha.bar” and “alpha/delta”.

Usage

ghyp(lambda = 0.5, chi = 0.5, psi = 2, mu = 0, sigma = diag(rep(1, length(mu))),
     gamma = rep(0, length(mu)), alpha.bar = NULL, data = NULL)

ghyp.ad(lambda = 0.5, alpha = 1.5, delta = 1, beta = rep(0, length(mu)),
        mu = 0, Delta = diag(rep(1, length(mu))), data = NULL)


hyp(chi = 0.5, psi = 2, mu = 0, sigma = diag(rep(1, length(mu))),
    gamma = rep(0, length(mu)), alpha.bar = NULL, data = NULL)

hyp.ad(alpha = 1.5, delta = 1, beta = rep(0, length(mu)), mu = 0,
       Delta = diag(rep(1, length(mu))), data = NULL)


NIG(chi = 2, psi = 2, mu = 0, sigma = diag(rep(1, length(mu))),
    gamma = rep(0, length(mu)), alpha.bar = NULL, data = NULL)

NIG.ad(alpha = 1.5, delta = 1, beta = rep(0, length(mu)), mu = 0,
       Delta = diag(rep(1, length(mu))), data = NULL)


student.t(nu = 3.5, chi = nu - 2, mu = 0, sigma = diag(rep(1, length(mu))),
          gamma = rep(0, length(mu)), data = NULL)

student.t.ad(lambda = -2, delta = 1, beta = rep(0, length(mu)), mu = 0,
             Delta = diag(rep(1, length(mu))), data = NULL)


VG(lambda = 1, psi = 2*lambda, mu = 0, sigma = diag(rep(1, length(mu))),
   gamma = rep(0, length(mu)), data = NULL)

VG.ad(lambda = 2, alpha = 1.5, beta = rep(0, length(mu)), mu = 0,
      Delta = diag(rep(1, length(mu))), data = NULL)

gauss(mu = 0, sigma = diag(rep(1, length(mu))), data = NULL)

Arguments

lambda	Shape parameter. Common for all parametrizations.
nu	Shape parameter only used in case of a Student-t distribution in the “chi/psi” and “alpha.bar” parametrization . It determines the degree of freedom.
chi	Shape parameter of the “chi/psi” parametrization.
psi	Shape parameter of the “chi/psi” parametrization.
alpha	Shape parameter of the “alpha/delta” parametrization.
delta	Shape parameter of the “alpha/delta” parametrization.
alpha.bar	Shape parameter of the “alpha.bar” parametrization. Supplying “alpha.bar” makes the parameters “chi” and “psi” redundant.
mu	Location parameter. Either a scalar or a vector. Common for all parametrizations.
sigma	Dispersion parameter of the “chi/psi” parametrization. Either a scalar or a matrix.
Delta	Dispersion parameter. Must be a matrix with a determinant of 1. This parameter is only used in the multivariate case of the “alpha.beta” parametrization.
gamma	Skewness parameter of the “chi/psi” parametrization. Either a scalar or a vector.
beta	Skewness parameter of the “alpha/delta” parametrization. Either a scalar or a vector.
data	An object coercible to a vector (univariate case) or matrix (multivariate case).

Details

These functions serve as constructors for univariate and multivariate objects.

ghyp, hyp and NIG are constructor functions for both the “chi/psi” and the “alpha.bar” parametrization. Whenever alpha.bar is not NULL it is assumed that the “alpha.bar” parametrization is used and the parameters “chi” and “psi” become redundant.

Similarly, the variance gamma (VG) and the Student-t distribution share the same constructor function for both the chi/psi and alpha.bar parametrization. To initialize them in the alpha.bar parametrization simply omit the argument psi and chi, respectively. If psi or chi are submitted, the “chi/psi” parametrization will be used.

ghyp.ad, hyp.ad, NIG.ad, student.t.ad and VG.ad use the “alpha/delta” parametrization.

The following table gives the constructors for each combination of distribution and parametrization.

		Parametrization
Distribution	“chi/psi”	“alpha.bar”	“alpha/delta”
GH	ghyp(...)	ghyp(..., alpha.bar=x)	ghyp.ad(...)
hyp	hyp(...)	hyp(..., alpha.bar=x)	hyp.ad(...)
NIG	NIG(...)	NIG(..., alpha.bar=x)	NIG.ad(...)
Student-t	student.t(..., chi=x)	student.t(...)	student.t.ad(...)
VG	VG(..., psi=x)	VG(...)	VG.ad(...)

Have a look on the vignette of this package in the doc folder for further information regarding the parametrization and for the domains of variation of the parameters.

Value

An object of class ghyp.

Note

The Student-t parametrization obtained via the “alpha.bar” parametrization slightly differs from the common Student-t parametrization: The parameter sigma denotes the standard deviation in the univariate case and the variance in the multivariate case. Thus, set \sigma = \sqrt{\nu /(\nu - 2)} in the univariate case to get the same results as with the standard R implementation of the Student-t distribution.

In case of non-finite variance, the “alpha.bar” parametrization does not work because sigma is defined to be the standard deviation. In this case the “chi/psi” parametrization can be used by submitting the parameter chi. To obtain equal results as the standard R implmentation use student.t(nu = nu, chi = nu) (see Examples).

Have a look on the vignette of this package in the doc folder for further information.

Once an object of class ghyp is created the methods Xghyp have to be used even when the distribution is a special case of the GH distribution. E.g. do not use dVG. Use dghyp and submit a variance gamma distribution created with VG().

Author(s)

David Luethi

References

ghyp-package vignette in the doc folder or on https://cran.r-project.org/package=ghyp

See Also

ghyp-class for a summary of generic methods assigned to ghyp objects, coef for switching between different parametrizations, d/p/q/r/ES/gyhp for density, distribution function et cetera, fit.ghypuv and fit.ghypmv for fitting routines.

Examples

  ## alpha.bar parametrization of a univariate GH distribution
  ghyp(lambda=2, alpha.bar=0.1, mu=0, sigma=1, gamma=0)
  ## lambda/chi parametrization of a univariate GH distribution
  ghyp(lambda=2, chi=1, psi=0.5, mu=0, sigma=1, gamma=0)
  ## alpha/delta parametrization of a univariate GH distribution
  ghyp.ad(lambda=2, alpha=0.5, delta=1, mu=0, beta=0)

  ## alpha.bar parametrization of a multivariate GH distribution
  ghyp(lambda=1, alpha.bar=0.1, mu=2:3, sigma=diag(1:2), gamma=0:1)
  ## lambda/chi parametrization of a multivariate GH distribution
  ghyp(lambda=1, chi=1, psi=0.5, mu=2:3, sigma=diag(1:2), gamma=0:1)
  ## alpha/delta parametrization of a multivariate GH distribution
  ghyp.ad(lambda=1, alpha=2.5, delta=1, mu=2:3, Delta=diag(c(1,1)), beta=0:1)

  ## alpha.bar parametrization of a univariate hyperbolic distribution
  hyp(alpha.bar=0.3, mu=1, sigma=0.1, gamma=0)
  ## lambda/chi parametrization of a univariate hyperbolic distribution
  hyp(chi=1, psi=2, mu=1, sigma=0.1, gamma=0)
  ## alpha/delta parametrization of a univariate hyperbolic distribution
  hyp.ad(alpha=0.5, delta=1, mu=0, beta=0)

  ## alpha.bar parametrization of a univariate NIG distribution
  NIG(alpha.bar=0.3, mu=1, sigma=0.1, gamma=0)
  ## lambda/chi parametrization of a univariate NIG distribution
  NIG(chi=1, psi=2, mu=1, sigma=0.1, gamma=0)
  ## alpha/delta parametrization of a univariate NIG distribution
  NIG.ad(alpha=0.5, delta=1, mu=0, beta=0)

  ## alpha.bar parametrization of a univariate VG distribution
  VG(lambda=2, mu=1, sigma=0.1, gamma=0)
  ## alpha/delta parametrization of a univariate VG distribution
  VG.ad(lambda=2, alpha=0.5, mu=0, beta=0)

  ## alpha.bar parametrization of a univariate t distribution
  student.t(nu = 3, mu=1, sigma=0.1, gamma=0)
  ## alpha/delta parametrization of a univariate t distribution
  student.t.ad(lambda=-2, delta=1, mu=0, beta=1)

  ## Obtain equal results as with the R-core parametrization
  ## of the t distribution:
  nu <- 4
  standard.R.chi.psi <- student.t(nu = nu, chi = nu)
  standard.R.alpha.bar <- student.t(nu = nu, sigma = sqrt(nu  /(nu - 2)))

  random.sample <- rnorm(3)
  dt(random.sample, nu)
  dghyp(random.sample, standard.R.chi.psi)   # all implementations yield...
  dghyp(random.sample, standard.R.alpha.bar) # ...the same values

  random.quantiles <- runif(4)
  qt(random.quantiles, nu)
  qghyp(random.quantiles, standard.R.chi.psi)   # all implementations yield...
  qghyp(random.quantiles, standard.R.alpha.bar) # ...the same values

  ## If nu <= 2 the "alpha.bar" parametrization does not exist, but the
  ## "chi/psi" parametrization. The case of a Cauchy distribution:
  nu <- 1
  standard.R.chi.psi <- student.t(nu = nu, chi = nu)

  dt(random.sample, nu)
  dghyp(random.sample, standard.R.chi.psi)   # both give the same result

  pt(random.sample, nu)
  pghyp(random.sample, standard.R.chi.psi) # both give the same result

ghyp documentation built on Sept. 12, 2024, 7:38 a.m.