mixbox (version 1.2.3)

rptstable: 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)

Value

simulated realizations of size \(n\) from ptstable distribution.

Arguments

n

size of required samples.

alpha

tail thickness parameter.

beta

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