rptstable | R Documentation |
\alpha
-stable (ptstable) random variable.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}}}.
rptstable(n, alpha, beta)
n |
size of required samples. |
alpha |
tail thickness parameter. |
beta |
tilting parameter. |
simulated realizations of size n
from ptstable distribution.
Mahdi Teimouri
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")}.
n <- 100
alpha <- 1.4
beta <- 0.5
rptstable(n, alpha, beta)
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