Probabilities and Quantiles

Peter Straka

2021-09-05

Introduction

This vignette details 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 Garrappa (2015).

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).

First type Mittag-Leffler distribution

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).

Second type Mittag-Leffler distribution

Meerschaert and Scheffler (2004) 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.

References

Garrappa, Roberto. 2015. “Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions.” SIAM J. Numer. Anal. 53 (3): 1350–69. https://doi.org/10.1137/140971191.

Haubold, H. J., A. M. Mathai, and R. K. Saxena. 2011. “Mittag-Leffler Functions and Their Applications.” J. Appl. Math. 2011: 1–51. https://doi.org/10.1155/2011/298628.

Meerschaert, Mark M, and Hans-Peter Scheffler. 2004. “Limit Theorems for Continuous-Time Random Walks with Infinite Mean Waiting Times.” J. Appl. Probab. 41 (3): 623–38. https://doi.org/10.1239/jap/1091543414.