Probabilities and Quantiles

Mittag-Leffler function

Write E_α, β(z) for the two-parameter Mittag-Leffler function, and E_α(z) := E_α, 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_α(−y^α)

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^α) (using the package stabledist) and X^1/α 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:

P[E ≤ q] = P[Y^−α ≤ q] = P[Y ≥ q^−1/α]

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))^−α, where p ∈ (0, 1) and F_Y⁻¹ denotes the quantile function of Y, implemented in stabledist. Then one confirms

F_Y(q^−1/α) = 1 − p ⇒ P[Y ≥ q^−1/α] = p ⇒ P[Y^−α ≤ q] = p

which means F_E(q) = p.

rml()

Mittag-Leffler random variables E of second type are directly simulated as Y^−α, using stabledist.