A (non‐central) chi‐squared mixture of non‐central chi‐squareds is (non‐central) chi‐squared and related results, corollaries and applications

Abstract

Our main, novel, result is that a certain non-central chi-squared mixture of noncentral chi-squared distributions is itself a scaled non-central chi-squared distribution.From this and a link to a known result on a mixture representation for a scaled central chi-squared distribution, numerous further mixture results, both old and new, ensue.These include mixture results for central F, non-central F and Libby-Novick distributions. The main result involves distributions all with the same degrees of freedom; it is also extended to the case where the mixing non-central chi-squared distribution has degrees of freedom an even number larger than that of the conditional noncentral chi-squared distribution, with further consequences pursued.

A slew of mixture relationships involving discrete and continuous generalized hypergeometric distributions and their special cases

“…This mixture relationship between beta and Libby–Novick distributions complements the transformation relationship between them. It can first be found in Chabot (2016); see Jones and Marchand (2021). The following novel result also arises from (6)–(8): It is also straightforward to show that if , then .…”

“…In addition, ( 25), ( 26) and ( 16) show the standard construction of the negative binomial distribution as a gamma mixture of Poissons. The duality of these relationships was noted in Jones and Marchand (2021). 21), ( 22) and ( 12) is, in itself, of little interest.…”

“…This mixture relationship between beta and Libby-Novick distributions complements the transformation relationship between them. It can first be found in Chabot (2016); see Jones and Marchand (2021). The following novel result also arises from ( 6)-( 8):…”