The problem of combining several independent Chi squared or $F$ tests is considered. The data consist of $n$ independent Chi squared or $F$ variables on which tests of the null hypothesis that all noncentrality parameters are zero are based. In each case, necessary conditions and sufficient conditions for a test to be admissible are given in terms of the monotonicity and convexity of the acceptance region. The admissibility or inadmissibility of several tests based upon the observed significance levels of the individual test statistics is determined. In the Chi squared case, Fisher's and Tippett's procedures are admissible, the inverse normal and inverse logistic procedures are inadmissible, and the test based upon the sum of the significance levels is inadmissible when the level is less than a half. The results are similar, but not identical, in the $F$ case. Several generalized Bayes tests are derived for each problem.
John I. Marden. "Combining Independent Noncentral Chi Squared or $F$ Tests." Ann. Statist. 10 (1) 266 - 277, March, 1982. https://doi.org/10.1214/aos/1176345709