Set parameters:

```
library(MittagLeffleR)
<- 0.7
tail <- 2
scale <- 1000
n <- 2 * scale cutoff
```

Generate 1000 i.i.d. random variables, and plot their empirical CDF (black) against the population CDF (red). The curved dashed line represents the stretched exponential function, due to the asymptotic result

\[f(x; \alpha, \tau) \sim \exp(-(x/\tau)^\alpha), \quad x \downarrow 0.\]

The straight dashed line in the second plot represents the function \(x^{-\alpha}\), showing asymptotic equivalence to a power-law for large values.

```
<- rml(n = n, tail = tail, scale=scale)
r <- ecdf(r)
edfun <- seq(0.01,10,0.01)
x plot(x,edfun(x), xlim=c(0,10), type='l', main = "CDF on linear scale",
ylab="p", xlab="x")
<- pml(q = x, tail = tail, scale=scale)
y lines(x,y,col=2)
<- 1-exp(-(x/scale)^tail)
z lines(x,z, lty=2)
```

```
<- exp(seq(-10,10,0.01))
x <- 1-edfun(x)
y plot(x,y, type='l', log='xy', main = "Tail Function on log-scale",
xlab = "x", ylab = "p")
<- pml(q = x, tail = tail, scale=scale, lower.tail = FALSE)
y lines(x,y, col=2)
# power law for large values
<- x^(-tail)
z lines(x,z, lty=2)
# stretched exponential for small values
<- exp(-(x/scale)^tail)
w lines(x,w, lty=2)
```

A plot of the density:

```
<- 10
cutoff <- sum(r <= cutoff) / n
fac <- r[r <= cutoff]
r hist(r, freq = FALSE, breaks = 50)
<- seq(0.01,cutoff,0.01)
x <- dml(x = x, tail = tail, scale=scale) / fac
y lines(x,y, col=2)
```

The second type of Mittag-Leffler distribution is light-tailed, and in fact has finite moments of all orders: it drops off faster than the exponential distribution (dashed line).

```
library(MittagLeffleR)
<- 10^5
n <- 0.6
tail <- rml(n = n, tail = tail, scale=scale, second.type = TRUE)
r <- ecdf(r)
edfun plot(edfun, xlim=c(0,cutoff))
<- seq(0.01,cutoff,0.01)
x <- pml(q = x, tail = tail, scale=scale, second.type = TRUE)
y lines(x,y, col=2)
```

```
<- exp(seq(-10,4,0.01))
x <- 1-edfun(x)
y plot(x,y, type='l', log='xy', main = "Tail Function on log-scale",
xlab = "x", ylab = "p")
<- pml(q = x, tail = tail, scale=scale, lower.tail = FALSE, second.type = TRUE)
y lines(x,y, col=2)
# exponential distribution
<- exp(-(x/scale))
w lines(x,w, lty=2)
```

A plot of the density:

```
hist(r, freq = FALSE, breaks = 20)
<- seq(0.01,cutoff,0.01)
x <- dml(x = x, tail = tail, scale=scale, second.type = TRUE)
y lines(x,y, col=2)
```