rptstable: Simulating from polynomially tilted alpha-stable (ptstable)...

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Simulating from polynomially tilted \alpha-stable (ptstable) random variable.

Description

Using method of Devroye (2009), we can generate from ptstable random variable. The density function of a ptstable distribution is given by

f_{T}(t|{\bold{\theta}})=\frac{\Gamma(1+\beta)}{\Gamma\Bigl(1+\frac{\beta}{\alpha}\Bigr)}t^{-\beta}f_{P}(t|\alpha),

where {\bold{\theta}}=(\alpha,\beta)^{\top} in which 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,

for

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}}}.

Usage

rptstable(n, alpha, beta)

Arguments

n	size of required samples.
alpha	tail thickness parameter.
beta	tilting parameter.

Value

simulated realizations of size n from ptstable distribution.

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

    n <- 100
alpha <- 1.4
beta  <- 0.5
rptstable(n, alpha, beta)

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