udmeixner function

UNU.RAN object for Meixner distribution

Create UNU.RAN object for a Meixner distribution with scale parameter alpha, asymmetry (shape) parameter beta, shape parameter delta and location parameter mu.

[Distribution] -- Meixner.

udmeixner(alpha, beta, delta, mu, lb=-Inf, ub=Inf)

Arguments

• alpha: scale parameter (must be strictly positive).
• beta: asymmetry (shape) parameter (must be larger than $-pi$ and smaller than $pi$).
• delta: shape parameter (must be strictly positive).
• mu: location parameter.
• lb: lower bound of (truncated) distribution.
• ub: upper bound of (truncated) distribution.

Details

The Mexiner distribution with parameters $alpha$, $beta$, $delta$, and $mu$

has density

$$f(x) = \kappa \,\exp(\beta(x-\mu)/\alpha)\, |\Gamma\left(\delta+ i(x-\mu)/\alpha\right)|^2f(x) = kappa * exp(beta*(x-mu)/alpha) * |Gamma(delta + i * (x-mu)/alpha)|^2$$

where the normalization constant is given by

$$\kappa =\frac{\left(2\cos\left(\beta/2\right)\right)^{2\delta}}{2 \alpha \pi \, \Gamma\left(2 \delta\right)}kappa = (2*cos(beta/2))^(2*delta) / (2 * alpha * pi * Gamma(2*delta))$$

The symbol $i$ denotes the imaginary unit, that is, we have to evaluate the gamma function $Gamma(z)$ for complex arguments $z = x + i*y$.

Notice that $alpha>0$, $|beta| < pi$

and $delta>0$.

The domain of the distribution can be truncated to the interval (lb,ub).

Returns

An object of class "unuran.cont".

See Also

unuran.cont.

References

Grigelionis, B., 1999. Processes of Meixner type. Lithuanian Mathematical Journal, Vol. 39, p. 33--41.

Schoutens, W., 2001. The Meixner Processes in Finance. Eurandom Report 2001-002, Eurandom, Eindhoven.

Author(s)

Josef Leydold and Kemal Dingec unuran@statmath.wu.ac.at .

Examples

## Create distribution object for meixner distribution
distr <- udmeixner(alpha=0.0298, beta=0.1271, delta=0.5729, mu=-0.0011)
## Generate generator object; use method PINV (inversion)
gen <- pinvd.new(distr)
## Draw a sample of size 100
x <- ur(gen,100)