mixbox (version 1.2.3)

dptstable: 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)

Value

The density function of polynomially tilted \(\alpha\)-stable distribution at point \(x\).

Arguments

x

point at which density value is desired.

param

tail thickness parameter.

Dim

tilting 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.

L. Devroye, (2009). Random variate generation for exponentially and polynomially tilted stable distributions, ACM Transactions on Modeling and Computer Simulation, 19(4), tools:::Rd_expr_doi("10.1145/1596519.1596523").

Examples

# \donttest{
    x <- 2
param <- 1.5
  Dim <- 2
dptstable(x, param, Dim)
# }