A (non‐central) chi‐squared mixture of non‐central chi‐squareds is (non‐central) chi‐squared and related results, corollaries and applications
M. C. Jones, Éric Marchand · Stat · 2021
Our main, novel, result is that a certain non‐central chi‐squared mixture of non‐central 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 centralF, non‐centralFand 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 non‐central chi‐squared distribution, with further consequences pursued.