Probabilities and Quantiles

Mittag-Leffler function

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 Haubold, Mathai, and Saxena (2011).

pml()

The cumulative distribution function at unit scale is (see Haubold, Mathai, and Saxena (2011))

$$F(y) = 1 - E_\alpha(-y^\alpha)$$

dml()

The probability density function at unit scale is (see Haubold, Mathai, and Saxena (2011))

$$f(y) = \frac{d}{dy} F(y) = y^{\alpha - 1} E_{\alpha, \alpha}(-y^\alpha)$$

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 Haubold, Mathai, and Saxena (2011).

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.