Mittag-Leffler distribution

The Mittag-Leffler distributions are two families of probability distributions on the half-line $[0,\infty)$. They are parametrized by a real $\alpha \in (0, 1]$ or $\alpha \in [0, 1]$. Both are defined with the Mittag-Leffler function, named after Gösta Mittag-Leffler.^[1]

The Mittag-Leffler function

For any complex $\alpha$ whose real part is positive, the series

$E_\alpha (z) := \sum_{n=0}^\infty \frac{z^n}{\Gamma(1+\alpha n)}$

defines an entire function. For $\alpha = 0$, the series converges only on a disc of radius one, but it can be analytically extended to $\mathbb{C} - \{1\}$.

First family of Mittag-Leffler distributions

The first family of Mittag-Leffler distributions is defined by a relation between the Mittag-Leffler function and their cumulative distribution functions.

For all $\alpha \in (0, 1]$, the function $E_\alpha$ is increasing on the real line, converges to $0$ in $- \infty$, and $E_\alpha (0) = 1$. Hence, the function $x \mapsto 1-E_\alpha (-x^\alpha)$ is the cumulative distribution function of a probability measure on the non-negative real numbers. The distribution thus defined, and any of its multiples, is called a Mittag-Leffler distribution of order $\alpha$.

All these probability distributions are Absolutely_continuous#Absolute_continuity_of_measures. Since $E_1$ is the exponential function, the Mittag-Leffler distribution of order $1$ is an exponential distribution. However, for $\alpha \in (0, 1)$, the Mittag-Leffler distributions are Heavy-tailed_distribution. Their Laplace transform is given by:

$\mathbb{E} (e^{- \lambda X_\alpha}) = \frac{1}{1+\lambda^\alpha},$

which implies that, for $\alpha \in (0, 1)$, the expectation is infinite. In addition, these distributions are geometric stable distributions.

Second family of Mittag-Leffler distributions

The second family of Mittag-Leffler distributions is defined by a relation between the Mittag-Leffler function and their moment-generating functions.

For all $\alpha \in [0, 1]$, a random variable $X_\alpha$ is said to follow a Mittag-Leffler distribution of order $\alpha$ if, for some constant $C>0$,

$\mathbb{E} (e^{z X_\alpha}) = E_\alpha (Cz),$

where the convergence stands for all $z$ in the complex plane if $\alpha \in (0, 1]$, and all $z$ in a disc of radius $1/C$ if $\alpha = 0$.

A Mittag-Leffler distribution of order $0$ is an exponential distribution. A Mittag-Leffler distribution of order $1/2$ is the distribution of the absolute value of a normal distribution random variable. A Mittag-Leffler distribution of order $1$ is a degenerate distribution. In opposition to the first family of Mittag-Leffler distribution, these distributions are not heavy-tailed.

These distributions are commonly found in relation with the local time of Markov processes. Parameter estimation procedures can be found here.^[2][3]

References

1. H. J. Haubold A. M. Mathai (2009). Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science: National Astronomical Observatory of Japan. Springer. p. 79. ISBN 978-3-642-03325-4.
2. D.O. Cahoy V.V. Uhaikin W.A. Woyczyński (2010). Parameter estimation for fractional Poisson processes. Journal of Statistical Planning and Inference 140. pp. 3106–3120.
3. D.O. Cahoy (2013). Estimation of Mittag-Leffler parameters. Communications in Statistics-Simulation and Computation. pp. 303–315.