I am a statistician and have no idea about the Meijer G-function. I am looking for a simple explanation of this function in any book or reference paper.
I don't have a book or a reference for it but would suggest a mode of thought that may appear "simple" to a statistician.
If start with a discrete model for choose n from N without replacement, then there are $C_n^N$ outcomes. If the population contains K particular items then the probability of drawing k of these within that n sample is called the hypergeometric distribution. You can write a table of recurrence relations in terms of factorial-like column factors (remaining proportion to find) weighted by a permutation that depends on the number of particular items remaining and the number of goes remaining.
This gives a 4-dimensional function on a discrete space of N,n,K and k with a lot of symmetry and we can consider z as an a-priori known quantity, the probability K/N.
Allowing z to vary independently from K/N is analogous to "uncertainty" in the remaining proportion having a dependence on the proportion found so far, the factorial-like series weights for Meijer are chosen to be a rational polynomial. This is a bit like Taylor series or moment-matching. I would imagine this as allowing the underlying Binomial to tend toward Gaussian smoothly.
Further generalising continuous coordinates in N,n,K and k requires an analytic "scaling law" that smoothly extends the function to a basis that passes through all these discrete points. Ensuring the resulting function family remains closed under differentiation and integration has practical use in terms of cumulative distributions etc., but I can't think of a realistic analogy (.. e.g. defining a "half coin toss" in terms of distribution convolutions can be done but is not physically meaningful or statistically interpretable even though the maths works)
From a Probability Theory point of view, it is nice to work with analytic functions on an infinite space with continuous smooth densities. Generalising the law of large numbers away from finite variance requires convolving higher order moments like leptokurtis or skew to some Levy alpha-stable distribution. It turns out that (for rational values of alpha) this can be written in terms of Meijer G-functions.
I don't have a book or a reference for it but would suggest a mode of thought that may appear "simple" to a statistician.
If start with a discrete model for choose n from N without replacement, then there are $C_n^N$ outcomes. If the population contains K particular items then the probability of drawing k of these within that n sample is called the hypergeometric distribution. You can write a table of recurrence relations in terms of factorial-like column factors (remaining proportion to find) weighted by a permutation that depends on the number of particular items remaining and the number of goes remaining.
This gives a 4-dimensional function on a discrete space of N,n,K and k with a lot of symmetry and we can consider z as an a-priori known quantity, the probability K/N.
Allowing z to vary independently from K/N is analogous to "uncertainty" in the remaining proportion having a dependence on the proportion found so far, the factorial-like series weights for Meijer are chosen to be a rational polynomial. This is a bit like Taylor series or moment-matching. I would imagine this as allowing the underlying Binomial to tend toward Gaussian smoothly.
Further generalising continuous coordinates in N,n,K and k requires an analytic "scaling law" that smoothly extends the function to a basis that passes through all these discrete points. Ensuring the resulting function family remains closed under differentiation and integration has practical use in terms of cumulative distributions etc., but I can't think of a realistic analogy (.. e.g. defining a "half coin toss" in terms of distribution convolutions can be done but is not physically meaningful or statistically interpretable even though the maths works)
From a Probability Theory point of view, it is nice to work with analytic functions on an infinite space with continuous smooth densities. Generalising the law of large numbers away from finite variance requires convolving higher order moments like leptokurtis or skew to some Levy alpha-stable distribution. It turns out that (for rational values of alpha) this can be written in terms of Meijer G-functions.
Thank you very much James. This was so good explanation. Regards..
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