| rpstable {mixSSG} | R Documentation |
Simulating positive stable random variable.
Description
The cumulative distribution function of positive stable distribution is given by
F_{P}(x)=\frac{1}{\pi}\int_{0}^{\pi}\exp\Bigl\{-x^{-\frac{\alpha}{2-\alpha}}a(\theta)\Bigr\}d\theta,
where 0<\alpha \leq 2 is tail thickness or index of stability and
a(\theta)=\frac{\sin\Bigl(\bigl(1-\frac{\alpha}{2}\bigr)\theta\Bigr)\Bigl[\sin \bigl(\frac{\alpha \theta}{2}\bigr)\Bigr]^{\frac{\alpha}{2-\alpha}}}{[\sin(\theta)]^{\frac{2}{2-\alpha}}}.
Kanter (1975) used the above integral transform to simulate positive stable random variable as
P\mathop=\limits^d\Bigl( \frac{a(\theta)}{W} \Bigr)^{\frac{2-\alpha}{\alpha}},
in which \theta\sim U(0,\pi) and W independently follows an exponential distribution with mean unity.
Usage
rpstable(n, alpha)Arguments
n |
the number of samples required. |
alpha |
the tail thickness parameter. |
Value
simulated realizations of size n from positive \alpha-stable distribution.
Author(s)
Mahdi Teimouri
References
M. Kanter, 1975. Stable densities under change of scale and total variation inequalities, Annals of Probability, 3(4), 697-707.
Examples
rpstable(10, alpha = 1.2)