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It is known (cf. Wikipedia, Noncentral_chi_distribution) that the non-central chi-square distribution with k degrees of freedom is a Poisson weighted mixture of central chi-squared distributions).

There are many proofs but all are basically the same (see same ref section "Derivation of the pdf" for a sketch and here libre stats for a freely accessible complete proof).

I'm looking for the intuition behind this result, that is, a way to interpret and explain it. Otherwise such the result seems kind of accident. Any idea ?

best, G.

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Patnaik writes on p. 203:

The [non-central $\chi^2$ distribution] has been [sic] obtained by [Fisher, R. A. (1928). Proc. Roy. Soc. A, 121, 654] as a particular case of the distribution of the multiple correlation coefficient. A purely analytical proof was given by [Tang, P. C. (1938). Statist. Res. Mem. 2, 126]. [...] We give a direct geometrical derivation of the [non-central $\chi^2$ distribution].

The geometric derivation by Patnaik uses a series expansion of the density, which is equivalent to the series expansion of the characteristic or moment generating function in the usual quick derivation of the non-central $\chi^2$ distribution as a Poisson mixture of central $\chi^2$ distributions.

So, I think there has never been any "intuition behind this result" -- just a lucky and rather simple analytical observation.

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  • $\begingroup$ Yes, I saw the paper and some others (for instance Kerridge Aust J Stat 7(1965)37-39 ) and there is no intuition behind just series expansion. But I still find weird that all constants line up so remarkably, My idea was to look for an intuition by decomposing somehow chi-square as a sum, the same for Poisson, both have additive properties... $\endgroup$
    – Gabriel
    Commented Feb 22, 2021 at 19:48
  • $\begingroup$ @Gabriel : Intuition is defined as the "faculty of knowing or understanding something without reasoning or proof" (thefreedictionary.com/intuition). It seems very likely that there has never been such intuition here. As for "all constants lin[ing] up so remarkably", I guess you mean the Poisson mixture. But the Poisson probabilities are just the terms of the normalized Maclaurin series for the exponential function, which are quite ubiquitous. $\endgroup$ Commented Feb 23, 2021 at 0:33
  • $\begingroup$ I guess you are right... $\endgroup$
    – Gabriel
    Commented Feb 24, 2021 at 16:25

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