Hyperbolic  R Documentation 
Density function, distribution function, quantiles and
random number generation for the hyperbolic distribution
with parameter vector Theta
. Utility routines are included for
the derivative of the density function and to find suitable break
points for use in determining the distribution function.
dhyperb(x, Theta, KNu = NULL, logPars = FALSE)
phyperb(q, Theta, small = 10^(6), tiny = 10^(10),
deriv = 0.3, subdivisions = 100, accuracy = FALSE, ...)
qhyperb(p, Theta, small = 10^(6), tiny = 10^(10),
deriv = 0.3, nInterpol = 100, subdivisions = 100, ...)
rhyperb(n, Theta)
ddhyperb(x, Theta, KNu = NULL, ...)
hyperbBreaks(Theta, small = 10^(6), tiny = 10^(10), deriv = 0.3, ...)
x,q 
Vector of quantiles. 
p 
Vector of probabilities. 
n 
Number of observations to be generated. 
Theta 
Parameter vector taking the form 
KNu 
Sets the value of the Bessel function in the density or derivative of the density. See Details 
.
logPars 
Logical; if 
small 
Size of a small difference between the distribution function and zero or one. See Details. 
tiny 
Size of a tiny difference between the distribution function and zero or one. See Details. 
deriv 
Value between 0 and 1. Determines the point where the derivative becomes substantial, compared to its maximum value. See Details. 
accuracy 
Uses accuracy calculated by 
subdivisions 
The maximum number of subdivisions used to integrate the density returning the distribution function. 
nInterpol 
The number of points used in qhyperb for cubic spline
interpolation (see 
... 
Passes arguments to 
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
BarndorffNielsen and Blæsild (1983). Three different
possible parameterisations 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
(\alpha,\beta)
(\phi,\gamma)
or
(\xi,\chi)
parameterisations to the
(\pi,\zeta)
parameterisation used above.
phyperb
breaks the real line into eight regions in order to
determine the integral of dhyperb
. The break points determining
the regions are found by hyperbBreaks
, based on the values of
small
, tiny
, and deriv
. In the extreme tails of
the distribution where the probability is tiny
according to
hyperbCalcRange
, the probability is taken to be zero. In the
range between where the probability is tiny
and small
according to hyperbCalcRange
, an exponential approximation to
the hyperbolic distribution is used. In the inner part of the
distribution, the range is divided in 4 regions, 2 above the mode, and
2 below. On each side of the mode, the break point which forms the 2
regions is where the derivative of the density function is
deriv
times the maximum value of the derivative on that side of
the mode. In each of the 4 inner regions the numerical integration routine
safeIntegrate
(which is a wrapper for
integrate
) is used to integrate the density dhyperb
.
qhyperb
uses the breakup of the real line into the same 8
regions as phyperb
. For quantiles which fall in the 2 extreme
regions, the quantile is returned as Inf
or Inf
as
appropriate. In the range between where the probability is tiny
and small
according to hyperbCalcRange
, an exponential
approximation to the hyperbolic distribution is used from which the
quantile may be found in closed form. In the 4 inner regions
splinefun
is used to fit values of the distribution function
generated by phyperb
. The quantiles are then found
using the uniroot
function.
phyperb
and qhyperb
may generally be expected to be
accurate to 5 decimal places.
The hyperbolic distribution is a special case of the generalized
hyperbolic distribution (BarndorffNielsen and Blæsild
(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).
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
.
hyperbBreaks
returns a list with components:
xTiny 
Value such that probability to the left is less than

xSmall 
Value such that probability to the left is less than

lowBreak 
Point to the left of the mode such that the
derivative of the density is 
highBreak 
Point to the right of the mode such that the
derivative of the density is 
xLarge 
Value such that probability to the right is less than

xHuge 
Value such that probability to the right is less than

modeDist 
The mode of the given hyperbolic distribution. 
David Scott d.scott@auckland.ac.nz, AiWei Lee, Jennifer Tso, Richard Trendall
BarndorffNielsen, 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.
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.
safeIntegrate
, integrate
for its
shortfalls, splinefun
, uniroot
and
hyperbChangePars
for changing parameters to the
(\pi,\zeta)
parameterisation, dghyp
for
the generalized hyperbolic distribution.
Theta < c(2,1,1,0)
hyperbRange < hyperbCalcRange(Theta, tol = 10^(3))
par(mfrow = c(1,2))
curve(dhyperb(x, Theta), from = hyperbRange[1], to = hyperbRange[2],
n = 1000)
title("Density of the\n Hyperbolic Distribution")
curve(phyperb(x, Theta), from = hyperbRange[1], to = hyperbRange[2],
n = 1000)
title("Distribution Function of the\n Hyperbolic Distribution")
dataVector < rhyperb(500, Theta)
curve(dhyperb(x, Theta), range(dataVector)[1], range(dataVector)[2],
n = 500)
hist(dataVector, freq = FALSE, add =TRUE)
title("Density and Histogram\n of the Hyperbolic Distribution")
logHist(dataVector, main =
"LogDensity and LogHistogram\n of the Hyperbolic Distribution")
curve(log(dhyperb(x, Theta)), add = TRUE,
range(dataVector)[1], range(dataVector)[2], n = 500)
par(mfrow = c(2,1))
curve(dhyperb(x, Theta), from = hyperbRange[1], to = hyperbRange[2],
n = 1000)
title("Density of the\n Hyperbolic Distribution")
curve(ddhyperb(x, Theta), from = hyperbRange[1], to = hyperbRange[2],
n = 1000)
title("Derivative of the Density\n of the Hyperbolic Distribution")
par(mfrow = c(1,1))
hyperbRange < hyperbCalcRange(Theta, tol = 10^(6))
curve(dhyperb(x, Theta), from = hyperbRange[1], to = hyperbRange[2],
n = 1000)
bks < hyperbBreaks(Theta)
abline(v = bks)
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