Macdonald distribution

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The three parameter Macdonald distribution, introduced by Nagar et al. [1] is defined by the probability density function (PDF):

macdonald distribution pdf

Where Γ, ν,σ is the extended gamma function, defined as:

extended gamma function

Other “Macdonald distribution”

Occasionally, the term “Macdonald distribution” is used to describe a theory from permutations (for example, see [2]), but this is a completely different concept from the distribution described by the above PDF.

Kropac [3] discusses distributions involving Macdonald functions (modified Bessel functions of the second kind), defined as:

Kn(z)= π(2 sin nπ)-1[I-n(z) – In(z)]

Johnson et. al [4] also refer to the Macdonald functions as distributions. One subclass of these functions containing functions with integer indices n are important in applications.


[1] D. K. Nagar, A. Roldán-Correa, and A. K. Gupta, “Extended matrix variate gamma and beta functions,” Journal of Multivariate Analysis, vol. 122, pp. 53–69, 2013.View at: Publisher Site | Google Scholar | MathSciNet

[2] Young, B. (2014). A Markov growth process for Macdonald’s distribution on reduced words. arXiv:1409.7714 (math).

[3] Kropac, O. (1982). Some properties and applications of probability distributions based on MacDonald function. Aplikace matematiky Volume: 27, Issue: 4, page 285-302 ISSN: 0862-7940

[4] Johnson, N. et al. Continuous Univariate Distributions. Wiley.