dptstable {mixbox} | R Documentation |
Monte Carlo approximation for density function of polynomially tilted alpha-stable distribution.
Description
The density function f_{T}(t|\alpha, \beta)
, of polynomially tilted \alpha
-stable distribution is given by (Devroye, 2009):
f_{T}(t | \alpha, \beta)=\frac{\Gamma(1+\beta)}{\Gamma\Bigl(1+\frac{\beta}{\alpha}\Bigr)}t^{-\beta}f_{P}(t|\alpha),
where 0<\alpha \leq 2
is tail thickness parameter or index of stability and \beta> 0
is tilting parameter. We note that f_{P}(t|\alpha)
is the density function of a positive \alpha
-stable distribution that has an integral representation (Kanter, 1975):
f_{P}(t|\alpha)=\frac{1}{\pi}\int_{0}^{\pi}{\frac{\alpha}{2-\alpha}}a(\theta) t^{-\frac{\alpha}{2-\alpha}-1}a(\theta) \exp\Bigl\{-t^{-\frac{\alpha}{2-\alpha}}a(\theta)\Bigr\}d\theta,
where
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
.
Usage
dptstable(x, param, Dim)
Arguments
x |
point at which density value is desired. |
param |
tail thickness parameter. |
Dim |
tilting parameter. |
Value
The density function of polynomially tilted \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.
L. Devroye, (2009). Random variate generation for exponentially and polynomially tilted stable distributions, ACM Transactions on Modeling and Computer Simulation, 19(4), doi: 10.1145/1596519.1596523.
Examples
x <- 2
param <- 1.5
Dim <- 2
dptstable(x, param, Dim)