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 Proof: (i) By (A.15) we have 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 41 IF ( z$_{0}$ , T$_{1}$ , ( B$_{0}$ , I )) = − ( ∂E$_{H}$$_{0}$W ( d ( B$_{0}$ , Σ$_{0}$ ) /σ$_{0}$ ) vec(( y − B ′ $_{0}$x ) x $^{′}$) ∂ vec( B $^{′}$)′ ) − 1 × W ( ‖ y$_{0}$ − B ′ $_{0}$x$_{0}$ ‖ /σ$_{0}$ ) vec(( y$_{0}$ − B ′ $_{0}$x$_{0}$ ) x ′ $_{0}$) . (A.27) E$_{H}$$_{0}$ ( ∂W ( d ( B$_{0}$ , Σ$_{0}$ ) /S )vec( ̂ u ( B$_{0}$ ) x $^{′}$) ∂ vec( Σ ) ′ ) = − E$_{H}$$_{0}$ ( W $^{′}$( ‖ u ‖ /S ) 2 S ‖ u ‖ ( xu ′ ⊗ I )( I ⊗ uu $^{′}$) ) ( ii ) E$_{H}$$_{0}$ ( W ( d ( B$_{0}$ , Σ$_{0}$ ) /S ) vec ( ̂ u ( B$_{0}$ ) x $^{′}$)) = 0 for all S. ( i ) E$_{H}$$_{0}$ ( ∂W ( d ( B$_{0}$ , Σ$_{0}$ ) /S ) vec ( ̂ u ( B$_{0}$ ) x $^{′}$) ∂ vec ( Σ ) ′ ) = 0 for all S,