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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 reference manual.


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)
#>      tail     scale    tailLo    tailHi   scaleLo   scaleHi 
#> 0.8998758 2.0170711 0.8995285 0.9002230 2.0151044 2.0190378

Read off

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|= t, ^2 = t)). A fractional diffusion at time (t) has the time-changed probability density

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

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

tail <- 0.65
dx <- 0.01
x <- seq(-2,5,dx)
t <- 3^(-1:2)
# cut off time so that only 1 % of probability is lost
umax <- qml(p = 0.99, tail = tail, scale = max(t), second.type = TRUE)
u <- seq(0.01,umax,dx)
H <- outer(u,t, function(u,t) {dml(x = u, tail = tail, scale = t^tail)})
N <- outer(x,u,function(x,u){dnorm(x = x, mean = u, sd = sqrt(u))})
p <- N %*% H * dx
df <- data.frame(p)
names(df) <- sapply(t, function(t){paste0("T=",round(t,2))})
df['x'] <- x
df %>% 
  gather(key = "time", value = "density", -x) %>%
  ggplot(mapping = aes(x=x, y=density, col=time)) + 
  geom_line() + 
  labs(ggtitle("Subdiffusion with drift"))


See the page for vignettes on