dpstable | R Documentation |
The density function f_{P}(p|\alpha)
, of positive \alpha
-stable distribution is given by (Kanter, 1975):
f_{P}(p|\alpha)=\frac{1}{\pi}\int_{0}^{\pi}{\frac{\alpha}{2-\alpha}}a(\theta) p^{-\frac{\alpha}{2-\alpha}-1}a(\theta) \exp\Bigl\{-p^{-\frac{\alpha}{2-\alpha}}a(\theta)\Bigr\}d\theta,
where 0<\alpha \leq 2
is tail thickness parameter 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}},}
for 0<\theta < \pi
. We use the Monte Carlo method for approximating f_{P}(p|\alpha)
.
dpstable(x, param)
x |
point at which density value is desired. |
param |
tail index parameter. |
The density function of positive \alpha
-stable distribution at point x
.
Mahdi Teimouri
M. Kanter, (1975). Stable densities under change of scale and total variation inequalities, Annals of Probability, 3(4), 697-707.
x <- 2
param <- 1.5
dpstable(x, param)
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