Sphericity test in a GMANOVA-MANOVA model with normal error

For the GMANOVA-MANOVA model with normal error: , , we study in this paper the sphericity hypothesis test problem with respect to covariance matrix: [Sigma]=[lambda]Iq ([lambda] is unknown). It is shown that, as a function of the likelihood ratio statistic [Lambda], the null distribution of [Lambda]2/n can be expressed by Meijer's function, and the asymptotic null distribution of -2log[Lambda] is (as n-->[infinity]). In addition, the Bartlett type correction -2[rho]log[Lambda] for log[Lambda] is indicated to be asymptotically distributed as with order n-2 for an appropriate Bartlett adjustment factor -2[rho] under null hypothesis.