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
.