Let ???? and be S-estimators of location and scatter based on the scale ????, and let

Let ????̂ and Σ̂ be S-estimators of location and scatter based on the scale ????̂, and let ????̂0 = ????̂(d(X, ̂????, Σ)) ̂ . Given a constant ????0, define ????̂∗ and Σ̂∗ as the values ????

and Σ that minimize |????| subject to ????̂(d(x, ????, Σ)) = ????0. Prove that ????̂∗ = ????̂ and

Σ̂∗ = (????̂0∕????0)Σ̂