content_image/297296.jpg content_image/297295.jpg content_image/297294.jpg content_image/297293.jpg
Proof of Proposition 7: Consider first the case Σ$_{0}$ = I$_{q}$ . The matrix Λ defined in (6.2) can also be expressed as 

<img src='content_image/297296.jpg'>

Since W$_{1}$ is differentiable with bounded derivative we can differentiating inside the expectation. We can now proceed analogously to the proof of (A.28) and we have that 

<img src='content_image/297295.jpg'>

Using the same arguments as before and W ( u ) = ψ$_{1}$ ( u ) /u , we obtain 

<img src='content_image/297294.jpg'>

Since ( E$_{G}$$_{0}$ xx ′ ⊗ I$_{q}$ ) − 1 = ( E$_{G}$$_{0}$ xx $^{′}$)− 1 ⊗ I$_{q}$ , the Proposition is proved for the case Σ$_{0}$ = I$_{q}$ . 

For the general case, let R a matrix such that Σ$_{0}$ = RR ′ and consider the following transformation y ∗ = R − $^{1}$y . Then B ∗ 0 = B$_{0}$R ′− 1 and y ∗ = B ∗ 0 $^{′}$x + u $^{∗}$, with u ∗ = R − $^{1}$u . Observe that the distribution of u ∗ is given by the density (4.1) with Σ$_{0}$ = I$_{q}$ and 

<img src='content_image/297293.jpg'>

and therefore, by the affine-equivariance of the MM-estimates, (6.4) follows. ■ 

