Sphericity test in a GMANOVA–MANOVA model with normal error

For the GMANOVA–MANOVA model with normal error: Y = X B 1 Z 1 ′ + B 2 Z 2 ′ + E , E ∼ N q × n ( 0 , I n ⊗ Σ ) , we study in this paper the sphericity hypothesis test problem with respect to covariance matrix: Σ = λ I q ( λ is unknown). It is shown that, as a function of the likelihood ratio statistic Λ , the null distribution of Λ 2 / n can be expressed by Meijer’s G q , q q , 0 function, and the asymptotic null distribution of − 2 log Λ is χ q ( q + 1 ) / 2 − 1 2 (as n → ∞ ). In addition, the Bartlett type correction − 2 ρ log Λ for log Λ is indicated to be asymptotically distributed as χ q ( q + 1 ) / 2 − 1 2 with order n − 2 for an appropriate Bartlett adjustment factor − 2 ρ under null hypothesis.