Mittag-Leffler family of distributions

Description

A generalization of the exponential distribution. Contains

• the Mittag-Leffler function mlf

• distributions (dml, pml, qml) and random variate generation (rml)

• a log-moment estimator (logMomentEstimator), and maximum likelihood estimator (mlmle)

Details

• Plots of the Mittag-Leffler distributions

• Details of Mittag-Leffler random variate generation

• Probabilities and Quantiles

Also see the package web page at https://strakaps.github.io/MittagLeffleR/reference/index.html

Author(s)

Maintainer: Peter Straka straka.ps@gmail.com

Authors:

• Katharina Hees heeskatharina@gmail.com

• Gurtek Gill

Other contributors:

• Roberto Garrappa roberto.garrappa@uniba.it [contributor]

See Also

Useful links:

• https://strakaps.github.io/MittagLeffleR/

• Report bugs at https://github.com/strakaps/MittagLeffleR/issues

Distribution functions and random number generation.

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

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

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; \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.

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, 1–51. 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

Examples

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)

Log-Moments Estimator for the Mittag-Leffler Distribution (Type 1).

Description

Tail and scale parameter of the Mittag-Leffler distribution are estimated by matching with the first two empirical log-moments (see Cahoy et al., doi: 10.1016/j.jspi.2010.04.016).

Usage

logMomentEstimator(x, alpha = 0.05)

Arguments

Value

A named vector with entries (nu, delta, nuLo, nuHi, deltaLo, deltaHi) where nu is the tail parameter and delta the scale parameter of the Mittag-Leffler distribution, with confidence intervals (nuLo, nuHi) resp. (deltaLo, deltaHi).

References

Cahoy, D. O., Uchaikin, V. V., & Woyczynski Wojbor, W. A. (2010). Parameter estimation for fractional Poisson processes. Journal of Statistical Planning and Inference, 140(11), 3106–3120. doi: 10.1016/j.jspi.2010.04.016

Cahoy, D. O. (2013). Estimation of Mittag-Leffler Parameters. Communications in Statistics - Simulation and Computation, 42(2), 303–315. doi: 10.1080/03610918.2011.640094

Examples

logMomentEstimator(rml(n = 1000, scale = 0.03, tail = 0.84), alpha=0.95)

Mittag-Leffler Function.

Description

The generalized (two-parameter) Mittag-Leffer function is defined by the power series

E_{\alpha,\beta} (z) = \sum_{k=0}^\infty z^k / \Gamma(\alpha k + \beta)

for complex z and complex \alpha, \beta with Real(\alpha) > 0 (only implemented for real valued parameters).

Usage

mlf(z, a, b = 1, g = 1)

Arguments

Value

mlf returns the value of the Mittag-Leffler function.

References

Garrappa, R. (2015). Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions. SIAM Journal on Numerical Analysis, 53(3), 1350–1369. doi: 10.1137/140971191

The Mittag-Leffler function. MathWorks File Exchange. https://au.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function

Examples

mlf(2,0.7)

Maximum Likelihood Estimation of the Mittag-Leffler distribution

Description

Optimizes the bivariate loglikelihood of the Mittag-Leffler distribution via optim. Uses logMomentEstimator for initial parameter values.

Usage

mlmle(data, ...)

Arguments

Value

The output of optim.

Examples

library(magrittr)
rml(n = 100, tail = 0.8, scale = 1000) %>% mlmle()