rpstable: 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)
Value
simulated realizations of size \(n\) from positive \(\alpha\)-stable distribution.
Arguments
n
the number of samples required.
alpha
the tail thickness parameter.
Author
Mahdi Teimouri
References
M. Kanter, 1975. Stable densities under change of scale and total variation inequalities, Annals of Probability, 3(4), 697-707.