Package 'MittagLeffleR'

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 [email protected]

Authors:

• Katharina Hees [email protected]

• Gurtek Gill

Other contributors:

• Roberto Garrappa [email protected] [contributor]

See Also

Useful links:

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

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

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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

x	A vector of non-negative data.
alpha	Confidence intervals are calculated at level 1 - alpha.

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)

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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

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.

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)

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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

z	The argument (real-valued)
a, b, g	Parameters of the Mittag-Leffler distribution; see Garrappa

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)

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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

data	Vector of class "numeric"
...	Additional parameters passed on to optim.

Value

The output of optim.

Examples

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

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