# MittagLeffleR: Distribution functions and random number generation. In MittagLeffleR: Mittag-Leffler Family of Distributions

## Description

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).

## Usage

 1 2 3 4 5 6 7 dml(x, tail, scale = 1, log = FALSE, second.type = FALSE) pml(q, tail, scale = 1, second.type = FALSE, lower.tail = TRUE, log.p = FALSE) qml(p, tail, scale = 1, second.type = FALSE, lower.tail = TRUE, log.p = FALSE) rml(n, tail, scale = 1, second.type = FALSE) 

## Arguments

 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[X ≤ x] otherwise, P[X > x] p vector of probabilities. n number of random draws.

## Details

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.

## Value

dml returns the density, pml returns the distribution function, qml returns the quantile function, and rml generates random variables.

## References

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

## Examples

 1 2 3 4 5 dml(1, 0.8) dml(1, 0.6, second.type=TRUE) pml(2, 0.7, 1.5) qml(p = c(0.25, 0.5, 0.75), tail = 0.6, scale = 100) rml(10, 0.7, 1) 

MittagLeffleR documentation built on Sept. 6, 2021, 9:11 a.m.