dpstable: Monte Carlo approximation for density function of positive...

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Monte Carlo approximation for density function of positive alpha-stable distribution.

Description

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).

Usage

dpstable(x, param)

Arguments

x	point at which density value is desired.
param	tail index parameter.

Value

The density function of positive \alpha-stable distribution at point x.

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

x <- 2
param <- 1.5
dpstable(x, param)

mixbox documentation built on May 29, 2024, 3:58 a.m.