dptstable: Monte Carlo approximation for density function of...

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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), \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1145/1596519.1596523")}.

Examples

    x <- 2
param <- 1.5
  Dim <- 2
dptstable(x, param, Dim)

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