Description Usage Arguments Details Value References Examples
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).
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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 ≤ x] otherwise, P[X > x] |
p |
vector of probabilities. |
n |
number of random draws. |
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; α, τ) = 1 - E_{α,1} (-(q/τ)^α)
for q ≥ 0, tail parameter 0 < α ≤ 1, and scale parameter τ > 0. Its PDF is given by
f(x; α, τ) = x^{α - 1} E_{α,α} [-(x/τ)^α] / τ^α.
As α approaches 1 from below, the Mittag-Leffler converges (weakly) to the exponential distribution. For 0 < α < 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^∞ \exp(-sx) f(x; α, 1) dx = E_{α,1}(-s)
It is light-tailed, i.e. all its moments are finite. At scale τ, its density is
f(x; α, τ) = f(x/τ; α, 1) / τ.
dml
returns the density,
pml
returns the distribution function,
qml
returns the quantile function, and
rml
generates random variables.
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
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