udvg | R Documentation |
Create UNU.RAN object for a Variance Gamma distribution
with shape parameter lambda
, shape parameter
alpha
, asymmetry (shape) parameter beta
, and
location parameter mu
.
[Distribution] – Variance Gamma.
udvg(lambda, alpha, beta, mu, lb=-Inf, ub=Inf)
lambda |
shape parameter (must be strictly positive). |
alpha |
shape parameter (must be strictly larger than
absolute value of |
beta |
asymmetry (shape) parameter. |
mu |
location parameter. |
lb |
lower bound of (truncated) distribution. |
ub |
upper bound of (truncated) distribution. |
The variance gamma distribution with parameters
\lambda
, \alpha
, \beta
, and \mu
has density
f(x) = \kappa \;
|x-\mu|^{\lambda-1/2}
\cdot \exp(\beta(x-\mu))
\cdot K_{\lambda-1/2}\left(\alpha|x-\mu|\right)
where the normalization constant is given by
\kappa =
\frac{\left(\alpha^2 - \beta^2\right)^{\lambda}}{
\sqrt{\pi} \, (2 \alpha)^{\lambda-1/2} \,
\Gamma\left(\lambda\right)}
K_{\lambda}(t)
is the modified Bessel function of
the third kind with index \lambda
.
\Gamma(t)
is the Gamma function.
Notice that \alpha>|\beta|
and
\lambda>0
.
The domain of the distribution can be truncated to the
interval (lb
,ub
).
An object of class "unuran.cont"
.
For \lambda \le 0.5
, the density has a pole at
\mu
.
Josef Leydold and Kemal Dingec unuran@statmath.wu.ac.at.
Eberlein, E., von Hammerstein, E. A., 2004. Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. In Seminar on Stochastic Analysis, Random Fields and Applications IV, Progress in Probability 58, R. C. Dalang, M. Dozzi, F. Russo (Eds.), Birkhauser Verlag, p. 221–264.
Madan, D. B., Seneta, E., 1990. The variance gamma (V.G.) model for share market returns. Journal of Business, Vol. 63, p. 511–524.
Raible, S., 2000. L\'evy Processes in Finance: Theory, Numerics, and Empirical Facts. Ph.D. thesis, University of Freiburg.
unuran.cont
.
## Create distribution object for variance gamma distribution
distr <- udvg(lambda=2.25, alpha=210.5, beta=-5.14, mu=0.00094)
## Generate generator object; use method PINV (inversion)
gen <- pinvd.new(distr)
## Draw a sample of size 100
x <- ur(gen,100)
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