vignettes/parametrisation.Rmd
parametrisation.RmdFor the efficient generation of random variates, we use the following useful fact (see e.g. Theorem 19.1 in Haubold, Mathai, and Saxena (2011)): A standard \(\alpha\)-Mittag-Leffler random variable \(Y\) has the representation:
\[Y \stackrel{d}{=} X^{1/\alpha} Z\]
where \(X\) is standard exponentially distributed, \(Z\) is \(\alpha\)-stable with Laplace Transform \[\mathbf E[\exp(-sZ)] = \exp(-s^\alpha),\] \(X\) and \(Z\) are independent, and \(\stackrel{d}{=}\) means equality in distribution.
n <- 5
x <- rexp(n)To generate such random variates \(Z\), we use
a <- 0.8
sigma <- (cos(pi*a/2))^(1/a)
z <- stabledist::rstable(n = n, alpha = a, beta = 1, gamma = sigma, delta = 0, pm = 1)Below are the details of the calculation. We use the parametrization of the stable distribution by Samorodnitsky and Taqqu (1994) as it has become standard. For \(\alpha \in (0,1)\) and \(\alpha \in (1,2)\),
\[\mathbf E[\exp(it Z)] = \exp\left\lbrace -\sigma^\alpha |t|^\alpha \left[1 - i \beta {\rm sgn}t \tan \frac{\pi \alpha}{2}\right] + i a t\right\rbrace\]
As in Meerschaert and Scheffler (2001), Equation (7.28), set
\[\sigma^\alpha = C \Gamma(1-\alpha) \cos \frac{\pi\alpha}{2},\]
for some constant \(C > 0\), set \(\beta = 1\), set \(a = 0\), and the log-characteristic function becomes
\[\begin{align} -C \frac{\Gamma(2-\alpha)}{1-\alpha} \cos \frac{\pi\alpha}{2} |t|^\alpha \left[1 - i\, {\rm sgn}(t) \tan \frac{\pi \alpha}{2}\right] \\ = -C \Gamma(1-\alpha)|t|^\alpha \left[ \cos \frac{\pi \alpha}{2} - i\,{\rm sgn}(t) \sin \frac{\pi \alpha}{2}\right] \\ = -C \Gamma(1-\alpha)|t|^\alpha\left(\exp(-i {\rm sgn}(t) \pi/2)\right)^\alpha \\ = -C \Gamma(1-\alpha)(-i |t| {\rm sgn}(t))^\alpha \\ = -C \Gamma(1-\alpha)(-it)^\alpha \end{align}\]
Setting \(t = is\) recovers the Laplace transform, and to match the Laplace transform \(\exp(-s^\alpha)\) of \(Z\), it is necessary that \(C \Gamma(1-\alpha) = 1\). But then \(\sigma^\alpha = \cos(\pi \alpha/2)\), and we see that
\[Z \sim S(\alpha, \beta, \sigma, a) = S(\alpha, 1, \cos(\pi\alpha/2)^{1/\alpha}, 0)\]