Description Usage Arguments Details Value References Examples

Probability density, cumulative distribution function, quantile function and random variate generation for the two types of Mittag-Leffler distribution. The Laplace inversion algorithm by Garrappa is used for the pdf and cdf (see https://www.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function).

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`x, q` |
vector of quantiles. |

`tail` |
tail parameter. |

`scale` |
scale parameter. |

`log, log.p` |
logical; if TRUE, probabilities p are given as log(p). |

`second.type` |
logical; if FALSE (default), first type of Mittag-Leffler distribution is assumed. |

`lower.tail` |
logical; if TRUE, probabilities are |

`p` |
vector of probabilities. |

`n` |
number of random draws. |

The Mittag-Leffler function `mlf`

defines two types of
probability distributions:

The **first type** of Mittag-Leffler distribution assumes the Mittag-Leffler
function as its tail function, so that the CDF is given by

*F(q; α, τ) = 1 - E_{α,1} (-(q/τ)^α)*

for *q ≥ 0*, tail parameter *0 < α ≤ 1*,
and scale parameter *τ > 0*.
Its PDF is given by

*f(x; α, τ) = x^{α - 1}
E_{α,α} [-(x/τ)^α] / τ^α.*

As *α* approaches 1 from below, the Mittag-Leffler converges
(weakly) to the exponential
distribution. For *0 < α < 1*, it is (very) heavy-tailed, i.e.
has infinite mean.

The **second type** of Mittag-Leffler distribution is defined via the
Laplace transform of its density f:

*\int_0^∞ \exp(-sx) f(x; α, 1) dx = E_{α,1}(-s)*

It is light-tailed, i.e. all its moments are finite.
At scale *τ*, its density is

*f(x; α, τ) = f(x/τ; α, 1) / τ.*

`dml`

returns the density,
`pml`

returns the distribution function,
`qml`

returns the quantile function, and
`rml`

generates random variables.

Haubold, H. J., Mathai, A. M., & Saxena, R. K. (2011). Mittag-Leffler Functions and Their Applications. Journal of Applied Mathematics, 2011, 1–51. doi: 10.1155/2011/298628

Mittag-Leffler distribution. (2017, May 3). In Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/w/index.php?title=Mittag-Leffler_distribution&oldid=778429885

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