Matrix variate p-value in MANOVA

The distribution functions of the matricvariate beta type I and II distributions are studied under real normed division algebras. The unified approach for real, complex, quaternions and octonions, also considers general properties and highlights the potential application of the exact emerging upper probabilities $P(\mathbf{B} > \mathbf{\Omega})$ and $P(\mathbf{F} > \mathbf{\nabla})$. In this setting, the matrix probabilities arise naturally as univariate extensions into the so termed matrix variate $p$-values. Then, a new criterion for the general multivariate linear hypothesis test can be proposed under a simple heuristic interpretation. The new technique can be applied in a number of classical statistical tests. In particular, the multivariate analysis of variance (MANOVA) is illustrated in two well known scenarios, and the performance of our exact method is compared with the existing approximated criteria.