This vignette explains how the functions dml()
, pml()
, qml()
and
rml()
are evaluated using the Mittag-Leffler function mlf()
and
functions from the package stabledist
.
Evaluation of the Mittag-Leffler function relies on the algorithm by
@Garrappa2015.
Write $E_{\alpha, \beta}(z)$ for the two-parameter Mittag-Leffler function, and $E_\alpha(z) := E_{\alpha, 1}(z)$ for the one-parameter Mittag-Leffler function. One has
$$E_{\alpha, \beta}(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\beta + \alpha k)}, \quad \alpha \in \mathbb C, \Re(\alpha) > 0, z \in \mathbb C,$$
see @Haubold2011a.
pml()
The cumulative distribution function at unit scale is (see @Haubold2011a)
$$F(y) = 1 - E_\alpha(-y^\alpha)$$
dml()
The probability density function at unit scale is (see @Haubold2011a)
$$f(y) = \frac{d}{dy} F(y) = y^{\alpha - 1} E_{\alpha, \alpha}(-y^\alpha)$$
qml()
The quantile function qml()
is calculated by numeric inversion of the cumulative
distribution function pml()
using stats::uniroot()
.
rml()
Mittag-Leffler random variables $Z$ are generated as the product of
a stable random variable $Y$ with Laplace Transform $\exp(-s^\alpha)$
(using the package stabledist
)
and $X^{1/\alpha}$ where $X$ is a unit exponentially distributed random
variable, see @Haubold2011a.
@limitCTRW introduce the inverse stable subordinator, a stochastic process $E(t)$. The random variable $E := E(1)$ has unit scale Mittag-Leffler distribution of second type, see the equation under Remark 3.1. By Corollary 3.1, $E$ is equal in distribution to $Y^{-\alpha}$:
$$E \stackrel{d}{=} Y^{-\alpha},$$
where $Y$ is a sum-stable randomvariable as above.
pml()
Using stabledist
, we can hence calculate the cumulative distribution function
of $E$:
$$\mathbf P[E \le q] = \mathbf P[Y^{-\alpha} \le q] = \mathbf P[Y \ge q^{-1/\alpha}]$$
dml()
The probability density function is evaluated using the formula
$$f(x) = \frac{1}{\alpha} x^{-1-1/\alpha} f_Y(x^{-1/\alpha})$$
where $f_Y(x)$ is the probability density of the stable random variable $Y$.
qml()
Let $q = (F_Y^{-1}(1-p))^{-\alpha}$, where $p \in (0,1)$ and
$F_Y^{-1}$ denotes the quantile function of $Y$, implemented in
stabledist
. Then one confirms
$$F_Y(q^{-1/\alpha}) = 1-p \Rightarrow \mathbf P[Y \ge q^{-1/\alpha}] = p \Rightarrow \mathbf P[Y^{-\alpha} \le q] = p$$
which means $F_E(q) = p$.
rml()
Mittag-Leffler random variables $E$ of second type are directly simulated
as $Y^{-\alpha}$, using stabledist
.
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