rpstable {mixSSG} | R Documentation |
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.
rpstable(n, alpha)
n |
the number of samples required. |
alpha |
the tail thickness parameter. |
simulated realizations of size n
from positive \alpha
-stable distribution.
Mahdi Teimouri
M. Kanter, 1975. Stable densities under change of scale and total variation inequalities, Annals of Probability, 3(4), 697-707.
rpstable(10, alpha = 1.2)