Mittag-Leffler Family of Distributions

Implements the Mittag-Leffler function, distribution, random variate generation, and estimation. Based on the Laplace-Inversion algorithm by Garrappa, R. (2015) .

The MittagLeffleR R package

The first type Mittag-Leffler distribution is a heavy-tailed distribution, and occurs mainly as a waiting time distribution in problems with “fractional” time scales, e.g. times between earthquakes.

The second type Mittag-Leffler distribution is light-tailed, and “inverse” to the sum-stable distributions. It typically models the number of events in fractional systems and is used for time-changes of stochastic processes, e.g. anomalous diffusion processes.


Stable release on CRAN

You can install MittagLeffleR from CRAN via


Development version on Github

Install the devtools package first, then

# install.packages("devtools")


See the reference manual at


Fitting a Mittag-Leffler distribution

Generate a dataset first:

y = rml(n = 10000, tail = 0.9, scale = 2)

Fit the distribution:

logMomentEstimator(y, 0.95)
#>        nu     delta      nuLo      nuHi   deltaLo   deltaHi 
#> 0.8979629 2.0015598 0.8976156 0.8983103 1.9996012 2.0035184

Read off

  • the shape parameter (0 < \nu < 1),
  • the scale parameter (\delta > 0),
  • their 95% confidence intervals.

Calculate the probability density of an anomalous diffusion process

Standard Brownian motion with drift (1) has, at time (t), has a normal probability density (n(x|\mu = t, \sigma^2 = t)). A fractional diffusion at time (t) has the time-changed probability density

[p(x,t) = \int n(x| \mu = u, \sigma^2 = u)h(u,t) du]

where (h(u,t)) is a second type Mittag-Leffler probability density with scale (t^\alpha). (We assume (t=1).)

tail <- 0.65
dx <- 0.01
x <- seq(-2,5,dx)
umax <- qml(p = 0.99, tail = tail, scale = 1, second.type = TRUE)
u <- seq(0.01,umax,dx)
h <- dml(x = u, tail = tail, scale = 1, second.type = TRUE)
N <- outer(x,u,function(x,u){dnorm(x = x, mean = u, sd = sqrt(u))})
p <- N %*% h * dx
plot(x,p, type='l', main = "Fractional diffusion with drift at t=1")


See the page for vignettes on

  • Plots of the Mittag-Leffler distributions
  • Details of Mittag-Leffler random variate generation
  • Probabilities and Quantiles


MittagLeffleR 0.3.0 [2018-04-24]

  • Added maximum likelihood estimator

MittagLeffleR 0.2.0 [2017-11-07]

  • Added log-moment estimator by Dexter Cahoy
  • Fixed links to Vignettes

MittagLeffleR 0.1.0 [2017-06-16]

  • First version on CRAN

Reference manual

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0.4.1 by Peter Straka, 5 months ago

Report a bug at

Browse source code at

Authors: Katharina Hees [aut] , Peter Straka [aut, cre] , Gurtek Gill [aut] , Roberto Garrappa [ctb]

Documentation:   PDF Manual  

Task views: Probability Distributions

GPL (>= 2) license

Imports stats, stabledist

Suggests knitr, rmarkdown, testthat, animation, magrittr

See at CRAN