Sphericity test in a GMANOVA–MANOVA model with normal error

For the GMANOVA–MANOVA model with normal error: Y=XB1Z1′+B2Z2′+E, E∼Nq×n(0,In⊗Σ)E∼Nq×n(0,In⊗Σ), we study in this paper the sphericity hypothesis test problem with respect to covariance matrix: Σ=λIqΣ=λIq (λλ is unknown). It is shown that, as a function of the likelihood ratio statistic ΛΛ, the null distribution of Λ2/nΛ2/n can be expressed by Meijer’s Gq,qq,0 function, and the asymptotic null distribution of −2logΛ−2logΛ is χq(q+1)/2−12 (as n→∞n→∞). In addition, the Bartlett type correction −2ρlogΛ−2ρlogΛ for logΛlogΛ is indicated to be asymptotically distributed as χq(q+1)/2−12 with order n−2n−2 for an appropriate Bartlett adjustment factor −2ρ−2ρ under null hypothesis.