Hyperbolic: Hyperbolic Distribution

Description

Density function, distribution function, quantiles and random number generation for the hyperbolic distribution with parameter vector param. Utility routines are included for the derivative of the density function and to find suitable break points for use in determining the distribution function.

Usage

dhyperb(x, mu = 0, delta = 1, alpha = 1, beta = 0,
        param = c(mu, delta, alpha, beta))
phyperb(q, mu = 0, delta = 1, alpha = 1, beta = 0,
        param = c(mu, delta, alpha, beta),
        lower.tail = TRUE, subdivisions = 100,
        intTol = .Machine$double.eps^0.25,
        valueOnly = TRUE, ...)
qhyperb(p, mu = 0, delta = 1, alpha = 1, beta = 0,
        param = c(mu, delta, alpha, beta),
        lower.tail = TRUE, method = c("spline", "integrate"),
        nInterpol = 501, uniTol = .Machine$double.eps^0.25,
        subdivisions = 100, intTol = uniTol, ...)
rhyperb(n, mu = 0, delta = 1, alpha = 1, beta = 0,
        param = c(mu, delta, alpha, beta))
ddhyperb(x, mu = 0, delta = 1, alpha = 1, beta = 0,
         param = c(mu, delta, alpha, beta))

Arguments

x,q

Vector of quantiles.

Vector of probabilities.

Number of observations to be generated.

$\mu$ is the location parameter. By default this is set to 0.

delta

$\delta$ is the scale parameter of the distribution. A default value of 1 has been set.

alpha

$\alpha$ is the tail parameter, with a default value of 1.

beta

$\beta$ is the skewness parameter, by default this is 0.

param

Parameter vector taking the form c(mu, delta, alpha, beta).

method

Character. If "spline" quantiles are found from a spline approximation to the distribution function. If "integrate", the distribution function used is always obtained by integration.

lower.tail

Logical. If lower.tail = TRUE, the cumulative density is taken from the lower tail.

subdivisions

The maximum number of subdivisions used to integrate the density and determine the accuracy of the distribution function calculation.

intTol

Value of rel.tol and hence abs.tol in calls to integrate. See integrate.

valueOnly

Logical. If valueOnly = TRUE calls to pghyp only return the value obtained for the integral. If valueOnly = FALSE an estimate of the accuracy of the numerical integration is also returned.

nInterpol

Number of points used in qghyp for cubic spline interpolation of the distribution function.

uniTol

Value of tol in calls to uniroot. See uniroot.

…

Passes arguments to uniroot. See Details.

Value

dhyperb gives the density, phyperb gives the distribution function, qhyperb gives the quantile function and rhyperb generates random variates. An estimate of the accuracy of the approximation to the distribution function may be found by setting accuracy = TRUE in the call to phyperb which then returns a list with components value and error.

ddhyperb gives the derivative of dhyperb.

Details

The hyperbolic distribution has density

$$f(x)=\frac{1}{2\delta\sqrt{1+\pi^2}K_1(\zeta)} % e^{-\zeta[\sqrt{1+\pi^2}\sqrt{1+(\frac{x-\mu}{\delta})^2}-% \pi\frac{x-\mu}{\delta}]}$$

where $K_1()$ is the modified Bessel function of the third kind with order 1.

A succinct description of the hyperbolic distribution is given in Barndorff-Nielsen and Bl<e6>sild (1983). Three different possible parameterizations are described in that paper. A fourth parameterization is given in Prause (1999). All use location and scale parameters $\mu$ and $\delta$. There are two other parameters in each case.

Use hyperbChangePars to convert from the $(\pi, \zeta)$ $(\phi, \gamma)$ or $(\xi, \chi)$ parameterizations to the $(\alpha, \beta)$ parameterization used above.

Each of the functions are wrapper functions for their equivalent generalized hyperbolic counterpart. For example, dhyperb calls dghyp. See dghyp.

The hyperbolic distribution is a special case of the generalized hyperbolic distribution (Barndorff-Nielsen and Bl<e6>sild (1983)). The generalized hyperbolic distribution can be represented as a particular mixture of the normal distribution where the mixing distribution is the generalized inverse Gaussian. rhyperb uses this representation to generate observations from the hyperbolic distribution. Generalized inverse Gaussian observations are obtained via the algorithm of Dagpunar (1989).

References

Barndorff-Nielsen, O. and Bl<e6>sild, 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.

Dagpunar, J.S. (1989). An easily implemented generalized inverse Gaussian generator Commun. Statist. -Simula., 18, 703--710.

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

Examples

Run this code

# NOT RUN {
param <- c(0, 2, 1, 0)
hyperbRange <- hyperbCalcRange(param = param, tol = 10^(-3))
par(mfrow = c(1, 2))
curve(dhyperb(x, param = param), from = hyperbRange[1], to = hyperbRange[2],
      n = 1000)
title("Density of the\n Hyperbolic Distribution")
curve(phyperb(x, param = param), from = hyperbRange[1], to = hyperbRange[2],
      n = 1000)
title("Distribution Function of the\n Hyperbolic Distribution")
dataVector <- rhyperb(500, param = param)
curve(dhyperb(x, param = param), range(dataVector)[1], range(dataVector)[2],
      n = 500)
hist(dataVector, freq = FALSE, add =TRUE)
title("Density and Histogram\n of the Hyperbolic Distribution")
DistributionUtils::logHist(dataVector, main =
        "Log-Density and Log-Histogram\n of the Hyperbolic Distribution")
curve(log(dhyperb(x, param = param)), add = TRUE,
      range(dataVector)[1], range(dataVector)[2], n = 500)
par(mfrow = c(2, 1))
curve(dhyperb(x, param = param), from = hyperbRange[1], to = hyperbRange[2],
      n = 1000)
title("Density of the\n Hyperbolic Distribution")
curve(ddhyperb(x, param = param), from = hyperbRange[1], to = hyperbRange[2],
      n = 1000)
title("Derivative of the Density\n of the Hyperbolic Distribution")
# }

Run the code above in your browser using DataLab