Macdonald distribution

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Three parameter Macdonald distribution

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

For the real part — Re(ν)>0 — and 𝜎 = 0, the extension of the gamma function reduces to the classical gamma function, Γ(ν; 0) = Γ(ν).

Other “Macdonald distribution”

Occasionally, the term “Macdonald distribution” is used to describe a theory from permutations (for example, see [2]), but this is a 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)]

The modified Bessel function K(x) is also sometimes called the Macdonald function.

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. The indices here refer to parameters of the Macdonald function. Macdonald functions with integer indices have several properties that make them highly suitable for applications such as orthogonal functions (orthogonal functions are two functions with an inner product of zero), which enable them to represent many different functions. Macdonald functions with integer indices also have a straightforward Fourier transform (a.k.a. the characteristic function), which simplifies their use in signal processing 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.

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