content_image/311383.jpg content_image/311382.jpg content_image/311381.jpg content_image/311380.jpg
By (A.16) and (A.17) and the equality vec( ux $^{′}$) = ( x ⊗ I$_{q}$ ) u we have that 

<img src='content_image/311383.jpg'>

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 

<img src='content_image/311382.jpg'>

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 

<img src='content_image/311381.jpg'>

<img src='content_image/311380.jpg'>

