By (A.16) and (A.17) and the equality vec( ux $^{′}$) = ( x ⊗ I$_{q}$ ) u we have that Since the distribution of u is assumed elliptical with Σ$_{0}$ = I , for any function h , E$_{F}$$_{0}$ ( h ( ‖ u ‖ ) u$_{i}$u$_{j}$ ) = 0 if i = j and E$_{F}$$_{0}$ ( h ( ‖ u ‖ ) u 2 $_{i}$) = E$_{F}$$_{0}$ ( h ( ‖ u ‖ ) ‖ u ‖ $^{2}$) /q . Then Combining (A.27) with (A.28) and using then matrix equality vec( C $^{′}$A ) = ( A ⊗ I )vec( C $^{′}$), we obtain the proof of the Theorem in the case Σ$_{0}$ = I . For the general case, let R be a matrix such that Σ$_{0}$ = RR ′ and consider the following transformation y ∗ = R − $^{1}$y . Then y ∗ = B ∗ 0 $^{′}$x + u $^{∗}$, with u ∗ = R − $^{1}$u y B ∗ 0 = B$_{0}$R ′− $^{1}$. Since the distribution of u ∗ is given by the density (4.1) with Σ$_{0}$ = I and by the affine-equivariance of the estimate, we have that 42 vec( B ′ $_{0}$) = vec( RB ∗ 0 $^{′}$) , IF ( z$_{0}$ , T$_{1}$ , B$_{0}$ , Σ$_{0}$ ) = R IF (( R − $^{1}$y$_{0}$ , x$_{0}$ ) , T$_{1}$ , B ∗ $_{0}$, I ) . ∂E$_{H}$$_{0}$W ( d ( B$_{0}$ , Σ$_{0}$ ) /σ$_{0}$ ) vec(( y − B ′ $_{0}$x ) x $^{′}$) ∂ vec( B $^{′}$)′ = − E$_{F}$$_{0}$W ′ ( ‖ u ‖ σ$_{0}$ ) ‖ u ‖ ( E$_{G}$$_{0}$ xx ′ ⊗ I ) σ$_{0}$ − E$_{F}$$_{0}$W ( ‖ u ‖ σ$_{0}$ ) ( E$_{G}$$_{0}$ xx ′ ⊗ I ) = − [ E$_{F}$$_{0}$W $^{′}$( ‖ u ‖ /σ$_{0}$ ) ‖ u ‖ σ$_{0}$ + E$_{F}$$_{0}$W ( ‖ u ‖ σ$_{0}$ )] ( E$_{G}$$_{0}$ xx ′ ⊗ I ) . (A.28) ∂E$_{H}$$_{0}$W ( d ( B$_{0}$ , Σ$_{0}$ ) /σ$_{0}$ ) vec(( y − B ′ $_{0}$x ) x $^{′}$) ∂ vec( B $^{′}$)′ = − W $^{′}$( d ( B$_{0}$ , σ 2 $_{0}$I$_{q}$ )) d ( B$_{0}$ , σ 2 $_{0}$I$_{q}$ ) σ 2 0 vec(( y − B ′ $_{0}$x ) x $^{′}$)( y − B ′ $_{0}$x ) $^{′}$( x ′ ⊗ I$_{q}$ ) − W ( d ( B$_{0}$ , σ 2 $_{0}$I$_{q}$ ) ) ( xx ′ ⊗ I$_{q}$ ) = − W ′ ‖ u ‖ σ$_{0}$ ) ( x ⊗ I$_{q}$ ) uu $^{′}$( x ′ ⊗ I$_{q}$ ) σ$_{0}$ ‖ u ‖ − W ( ‖ u ‖ σ$_{0}$ ) ( xx ′ ⊗ I$_{q}$ ) .