GeneralizedHyperbolic (version 0.8-6)

hyperbChangePars: Change Parameterizations of the Hyperbolic Distribution

Description

This function interchanges between the following 4 parameterizations of the hyperbolic distribution:

1. \(\mu, \delta, \pi, \zeta\)

2. \(\mu, \delta, \alpha, \beta\)

3. \(\mu, \delta, \phi, \gamma\)

4. \(\mu, \delta, \xi, \chi\)

The first three are given in Barndorff-Nielsen and Blæsild (1983), and the fourth in Prause (1999)

Usage

hyperbChangePars(from, to, param, noNames = FALSE)

Value

A numerical vector of length 4 representing param in the

to parameterization.

Arguments

from

The set of parameters to change from.

to

The set of parameters to change to.

param

"from" parameter vector consisting of 4 numerical elements.

noNames

Logical. When TRUE, suppresses the parameter names in the output.

Author

David Scott , Jennifer Tso, Richard Trendall

Details

In the 4 parameterizations, the following must be positive:

1. \(\zeta, \delta\)

2. \(\alpha, \delta\)

3. \(\phi, \gamma, \delta\)

4. \(\xi, \delta\)

Furthermore, note that in the second parameterization \(\alpha\) must be greater than the absolute value of \(\beta\), while in the fourth parameterization, \(\xi\) must be less than one, and the absolute value of \(\chi\) must be less than \(\xi\).

References

Barndorff-Nielsen, O. and Blæsild, P. (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S. and Read, C. B., Vol. 3, pp. 700--707. New York: Wiley.

Prause, K. (1999) The generalized hyperbolic models: Estimation, financial derivatives and risk measurement. PhD Thesis, Mathematics Faculty, University of Freiburg.

See Also

dhyperb

Examples

param1 <- c(2, 1, 3, 1)                    # Parameterization 1
param2 <- hyperbChangePars(1, 2, param1)   # Convert to parameterization 2
param2                                     # Parameterization 2
hyperbChangePars(2, 1, param2)             # Back to parameterization 1