Sphericity test in a GMANOVA-MANOVA model with normal error

For the GMANOVA-MANOVA model with normal error: Y=XB"1Z"1^'+B"2Z"2^'+E, E~N"q"x"n(0,I"[email protected][email protected]), we study in this paper the sphericity hypothesis test problem with respect to covariance matrix: @[email protected]"q (@l is unknown). It is shown that, as a function of the likelihood ratio statistic @L, the null distribution of @L^2^/^n can be expressed by Meijer's G"q","q^q^,^0 function, and the asymptotic null distribution of [email protected] is @g"q"("q"+"1")"/"2"-"1^2 (as n->~). In addition, the Bartlett type correction [email protected]@L for [email protected] is indicated to be asymptotically distributed as @g"q"("q"+"1")"/"2"-"1^2 with order n^-^2 for an appropriate Bartlett adjustment factor [email protected] under null hypothesis.