Let {(x1, y1), . . . .,(xn, yn)} be a regression dataset, and ????

̂ an S-estimator with finite BP equal to ????∗. Let D ⊂ (1, .., n) with #(D) < n????∗. Show that there exists K such that:

(a) ????

̂ as a function of the yi is constant if the yi with i ∉ D remain fixed and those with i ∈ D are changed in any way such that |yi| ≥ K.

(b) there exists ????̂ depending only on D such that ????

̂ verifies

∑

i∉D

????

(

ri(????

̂)

????̂

)

= min.

Then:

(c) Discuss why property

(a) does not mean that the that the value of the estimator is the same as if we omit the points (xi, yi) with i ∈ D.

(d) Show that properties (a)–

(c) also hold for MM-estimators.