Mittag-Leffler distribution

The Mittag-Leffler distributions are two families of probability distributions on the half-line . They are parametrized by a real or . Both are defined with the Mittag-Leffler function, named after Gösta Mittag-Leffler.[1]

The Mittag-Leffler function

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For any complex   whose real part is positive, the series

 

defines an entire function. For  , the series converges only on a disc of radius one, but it can be analytically extended to  .

First family of Mittag-Leffler distributions

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

For all  , the function   is increasing on the real line, converges to   in  , and  . Hence, the function   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  .

All these probability distributions are absolutely continuous. Since   is the exponential function, the Mittag-Leffler distribution of order   is an exponential distribution. However, for  , the Mittag-Leffler distributions are heavy-tailed, with

 

Their Laplace transform is given by:

 

which implies that, for  , the expectation is infinite. In addition, these distributions are geometric stable distributions. Parameter estimation procedures can be found here.[2][3]

Second family of Mittag-Leffler distributions

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

For all  , a random variable   is said to follow a Mittag-Leffler distribution of order   if, for some constant  ,

 

where the convergence stands for all   in the complex plane if  , and all   in a disc of radius   if  .

A Mittag-Leffler distribution of order   is an exponential distribution. A Mittag-Leffler distribution of order   is the distribution of the absolute value of a normal distribution random variable. A Mittag-Leffler distribution of order   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.

References

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  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. Astrophysics and Space Science Proceedings. 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 (11): 3106–3120. arXiv:1806.02774. doi:10.1016/j.jspi.2010.04.016.
  3. ^ D.O. Cahoy (2013). "Estimation of Mittag-Leffler parameters". Communications in Statistics - Simulation and Computation. 42 (2): 303–315. arXiv:1806.02792. doi:10.1080/03610918.2011.640094.