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).
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)
| 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 \(P[X \le x]\) otherwise, \(P[X > x]\) |
| p | vector of probabilities. |
| n | number of random draws. |
dml returns the density,
pml returns the distribution function,
qml returns the quantile function, and
rml generates random variables.
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 $$F(q; \alpha, \tau) = 1 - E_{\alpha,1} (-(q/\tau)^\alpha)$$ for \(q \ge 0\), tail parameter \(0 < \alpha \le 1\), and scale parameter \(\tau > 0\). Its PDF is given by $$f(x; \alpha, \tau) = x^{\alpha - 1} E_{\alpha,\alpha} [-(x/\tau)^\alpha] / \tau^\alpha.$$ As \(\alpha\) approaches 1 from below, the Mittag-Leffler converges (weakly) to the exponential distribution. For \(0 < \alpha < 1\), 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: $$\int_0^\infty \exp(-sx) f(x; \alpha, 1) dx = E_{\alpha,1}(-s)$$ It is light-tailed, i.e. all its moments are finite. At scale \(\tau\), its density is $$f(x; \alpha, \tau) = f(x/\tau; \alpha, 1) / \tau.$$
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
dml(1, 0.8)#> [1] 0.2557438dml(1, 0.6, second.type=TRUE)#> [1] 0.4832354pml(2, 0.7, 1.5)#> [1] 0.6597308#> [1] 11.64327 60.56854 284.67928rml(10, 0.7, 1)#> [1] 0.795198647 0.408205091 0.161459719 1.067977006 0.003716904 0.126989515 #> [7] 0.163714569 1.936521213 0.363758400 0.381153862