GeneralizedHyperbolic  R Documentation 
Density function, distribution function, quantiles and
random number generation for the generalized 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.
dghyp(x, Theta) pghyp(q, Theta, small = 10^(6), tiny = 10^(10), deriv = 0.3, subdivisions = 100, accuracy = FALSE, ...) qghyp(p, Theta, small = 10^(6), tiny = 10^(10), deriv = 0.3, nInterpol = 100, subdivisions = 100, ...) rghyp(n, Theta) ddghyp(x, Theta) ghypBreaks(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

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 qghyp for cubic spline
interpolation (see 
... 
Passes arguments to 
The generalized hyperbolic distribution has density
f(x)=c(lambda,alpha,beta,delta) (K_(lambda1/2)(alpha sqrt(delta^2+(xmu)^2)))/ ((sqrt(delta^2+(xmu)^2)/alpha)^(1/2lambda)) exp(beta(xmu))
where K_nu() is the modified Bessel function of the third kind with order nu, and
c(lambda,alpha,beta,delta)= (sqrt(alpha^2beta^2)/delta)^lambda/ (sqrt(2π)K_lambda(delta sqrt(alpha^2beta^2)))
Use ghypChangePars
to convert from the
(zeta, rho),
xi,chi), or
(alpha bar, beta bar) parameterisations
to the (alpha, beta) parameterisation used
above.
pghyp
breaks the real line into eight regions in order to
determine the integral of dghyp
. The break points determining
the regions are found by ghypBreaks
, based on the values of
small
, tiny
, and deriv
. In the extreme tails of
the distribution where the probability is tiny
according to
ghypCalcRange
, the probability is taken to be zero. In the
inner part of the distribution, the range is divided in 6 regions, 3
above the mode, and 3 below. On each side of the mode, there are two
break points giving the required three regions. The outer break point
is where the probability in the tail has the value given by the variable
small
. The inner break point 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 6 inner regions
the numerical integration routine safeIntegrate
(which
is a wrapper for integrate
) is used to integrate the
density dghyp
.
qghyp
use the breakup of the real line into the same 8
regions as pghyp
. For quantiles which fall in the 2 extreme
regions, the quantile is returned as Inf
or Inf
as
appropriate. In the 6 inner regions splinefun
is used to fit
values of the distribution function generated by pghyp
. The
quantiles are then found using the uniroot
function.
pghyp
and qghyp
may generally be expected to be
accurate to 5 decimal places.
The generalized hyperbolic distribution is discussed in Bibby and
S<f6>renson (2003). It can be represented as a particular
mixture of the normal distribution where the mixing distribution is the
generalized inverse Gaussian. rghyp
uses this representation
to generate observations from the generalized hyperbolic
distribution. Generalized inverse Gaussian observations are obtained
via the algorithm of Dagpunar (1989) which is implemented in
rgig
.
dghyp
gives the density, pghyp
gives the distribution
function, qghyp
gives the quantile function and rghyp
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 pghyp
which then returns
a list with components value
and error
.
ddghyp
gives the derivative of dghyp
.
ghypBreaks
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 generalized hyperbolic distribution. 
David Scott d.scott@auckland.ac.nz, Richard Trendall
BarndorffNielsen, 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.
Bibby, B. M. and S<f6>renson, M. (2003). Hyperbolic processes in finance. In Handbook of Heavy Tailed Distributions in Finance, ed., Rachev, S. T. pp. 212–248. Elsevier Science B.~V.
Dagpunar, J.S. (1989). An easily implemented generalised 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.
dhyperb
for the hyperbolic distribution,
dgig
for the generalized inverse Gaussian distribution
safeIntegrate
, integrate
for its
shortfalls, splinefun
,
uniroot
and ghypChangePars
for
changing parameters to the (alpha,beta)
parameterisation
Theta < c(1/2,3,1,1,0) ghypRange < ghypCalcRange(Theta, tol = 10^(3)) par(mfrow = c(1,2)) curve(dghyp(x, Theta), from = ghypRange[1], to = ghypRange[2], n = 1000) title("Density of the\n Generalized Hyperbolic Distribution") curve(pghyp(x, Theta), from = ghypRange[1], to = ghypRange[2], n = 1000) title("Distribution Function of the\n Generalized Hyperbolic Distribution") dataVector < rghyp(500, Theta) curve(dghyp(x, Theta), range(dataVector)[1], range(dataVector)[2], n = 500) hist(dataVector, freq = FALSE, add =TRUE) title("Density and Histogram of the\n Generalized Hyperbolic Distribution") logHist(dataVector, main = "LogDensity and LogHistogramof the\n Generalized Hyperbolic Distribution") curve(log(dghyp(x, Theta)), add = TRUE, range(dataVector)[1], range(dataVector)[2], n = 500) par(mfrow = c(2,1)) curve(dghyp(x, Theta), from = ghypRange[1], to = ghypRange[2], n = 1000) title("Density of the\n Generalized Hyperbolic Distribution") curve(ddghyp(x, Theta), from = ghypRange[1], to = ghypRange[2], n = 1000) title("Derivative of the Density of the\n Generalized Hyperbolic Distribution") par(mfrow = c(1,1)) ghypRange < ghypCalcRange(Theta, tol = 10^(6)) curve(dghyp(x, Theta), from = ghypRange[1], to = ghypRange[2], n = 1000) bks < ghypBreaks(Theta) abline(v = bks)
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