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

Jones, M. C. and Marchand, Éric (2021). A (non-central) chi-squared mixture of non-central chi-squareds is (non-central) chi-squared and related results, corollaries and applications. Stat, 10(1), article no. e398.

DOI: https://doi.org/10.1002/sta4.398

Abstract

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 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 non-central chi-squared distribution, with further consequences pursued.

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