■ Before proving Theorem 5 we need to prove several auxiliary Lemmas. For simplicity we will assume that the initial estimator ˜ B$_{n}$ is regression- and affine- equivariant and ˜ Σ$_{n}$ is affine-equivariant and regression-invariant. Then without loss of generality we can assume, due to Remark 3, that B$_{0}$ = 0 and Σ$_{0}$ = I . These assumptions are not essential for the proofs. Lemma 14. Let ( y ′ $_{i}$, x ′ $_{i}$) , 1 ≤ i ≤ n , be a random sample of the model (1.1) with parameters B$_{0}$ and Σ$_{0}$ , where the x$_{i}$ are random and Σ$_{0}$ = | Σ$_{0}$ | 1 $^{/q}$Γ$_{0}$ , and let ρ$_{0}$ be a ρ -function. Assume that the initial estimates ˜ B$_{n}$ and ˜ Σ$_{n}$ are consistent for B$_{0}$ and Γ$_{0}$ respectively; then ˆ σ$_{n}$ is consistent to σ$_{0}$ defined by the equation (5.1) . Proof: Take ε > 0, then by Lemma 9, we can find δ > 0 such that and where E = { ( B , Σ ) ∈ R p × q × S q : ‖ B ‖ ≤ δ, ‖ Σ − I$_{q}$ ‖ ≤ δ } . By the law of large numbers we have and 43 lim n −→∞ 1 n n ∑ i =1 inf E ρ$_{0}$ ( d$_{i}$ ( B , Σ ) ( σ$_{0}$ − ε ) ) ≥ b + δ a.s. lim n −→∞ 1 n n ∑ i =1 inf E ρ$_{0}$ ( d$_{i}$ ( B , Σ ) ( σ$_{0}$ + ε ) ) ≤ b − δ a.s.. E$_{H}$$_{0}$ ( inf E ρ$_{0}$ ( ( ( y − B $^{′}$x ) $^{′}$Σ− $^{1}$( y − B $^{′}$x ) ) 1 / 2 / ( σ$_{0}$ + ε ) )) ≤ b − δ E$_{H}$$_{0}$ ( inf E ρ$_{0}$ ( ( ( y − B $^{′}$x ) $^{′}$Σ− $^{1}$( y − B $^{′}$x ) ) 1 / 2 / ( σ$_{0}$ − ε ) )) ≥ b + δ