content_image/324225.jpg content_image/324224.jpg content_image/324223.jpg content_image/324222.jpg
Lemma 13. Consider the same assumptions of Theorem 4 and suppose that Σ$_{0}$ = I . Then, if H$_{0}$ is the distribution of ( y $^{′}$, x $^{′}$)$^{′}$, we have that 

<img src='content_image/324225.jpg'>

<img src='content_image/324224.jpg'>

Proof: 

<img src='content_image/324223.jpg'>

Since the distribution of u is assumed elliptical with Σ$_{0}$ = I , for any function h we have, E$_{H}$$_{0}$ ( h ( ‖ u ‖ ) u$_{i}$u$_{j}$u$_{l}$x$_{k}$ ) = 0. Then, since all the elements of the right side of the above equation have this form, part (i) of the lema is proved. (ii) follows from E$_{H}$$_{0}$ ( h ( ‖ u ‖ ) u$_{i}$x$_{j}$ ) = 0 for all i and j . ■ 

Proof Theorem 4: Assume z = ( y $^{′}$, x $^{′}$)′ satisfying the MLM (1.1). Consider first the case with Σ$_{0}$ = I . Using Lemma 11 with θ$_{1}$ = vec( B ′ $_{0}$), θ$_{2}$ = vec( Σ$_{0}$ ), S ( T$_{0}$ ( H ) , H ) = S ( H ) and lemmas 12 and 13 we obtained 

<img src='content_image/324222.jpg'>

