First type of Mittag-Leffler distribution
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.
r <- rml(n = n, tail = tail, scale=scale)
edfun <- ecdf(r)
x <- seq(0.01,10,0.01)
plot(x,edfun(x), xlim=c(0,10), type='l', main = "CDF on linear scale",
ylab="p", xlab="x")
y <- pml(q = x, tail = tail, scale=scale)
lines(x,y,col=2)
z <- 1-exp(-(x/scale)^tail)
lines(x,z, lty=2)
x <- exp(seq(-10,10,0.01))
y <- 1-edfun(x)
plot(x,y, type='l', log='xy', main = "Tail Function on log-scale",
xlab = "x", ylab = "p")
y <- pml(q = x, tail = tail, scale=scale, lower.tail = FALSE)
lines(x,y, col=2)
# power law for large values
z <- x^(-tail)
lines(x,z, lty=2)
# stretched exponential for small values
w <- exp(-(x/scale)^tail)
lines(x,w, lty=2)
A plot of the density:
cutoff <- 10
fac <- sum(r <= cutoff) / n
r <- r[r <= cutoff]
hist(r, freq = FALSE, breaks = 50)
x <- seq(0.01,cutoff,0.01)
y <- dml(x = x, tail = tail, scale=scale) / fac
lines(x,y, col=2)
