The Necessary and Sufficient Conditions for the Ratios of Quadratic Forms to be Distributed as F-distributions

The necessary and sufficient conditions such that the ratios of quardratic forms in multinor-mal random vectors have F-distributions are obtained by investigating the density functions and moments of F-distributions and moments of F-distributions. Furthermore, the necessary and sufficient conditions for the ratios of quadratic forms contoured random vectors to be distributed as F-distributions are also obtained. As applications, we extend the classical F-test to some dependent two group samples.