MittagLeffleR: Distribution functions and random number generation.

Description Usage Arguments Details Value References Examples

Description

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).

Usage

1
2
3
4
5
6
7
8
9
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)

Arguments

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 observations. If length(n) > 1, the length is taken to be the number required.

Details

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.

Value

dml returns the density, pml returns the distribution function, qml returns the quantile function, and rml generates random variables.

References

Haubold, H. J., Mathai, A. M., & Saxena, R. K. (2011). Mittag-Leffler Functions and Their Applications. Journal of Applied Mathematics, 2011, 1<e2><80><93>51. http://doi.org/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

Examples

1
2
3
4
5
dml(1, 0.8)
dml(1, 0.6, second.type=TRUE)
pml(2, 0.7, 1.5)
qml(p = c(0.25, 0.5, 0.75), tail = 0.6, scale = 100)
rml(10, 0.7, 1)

MittagLeffleR documentation built on May 2, 2019, 1:42 p.m.