For the model in Problem 7.3, define an estimator by the equation

∑n i=1

(yi − p(xi, ????))sgn(xi) = 0.

Since deleting all xi = 0 yields the same estimate, it will be henceforth assumed that xi ≠ 0 for all i.

(a) Show that this estimator is Fisher-consistent.

(b) Show that the estimator is a weighted ML estimator.

(c) Given the sample Zn = {(xi, yi), i = 1,..., n}, define the sample Z∗

n =

{(x∗

i , y∗

i ), i = 1,..., n}, where (x∗

i , y∗

i )=(xi, yi) if xi > 0 and (x∗

i , y∗

i ) =

(−xi, 1 − yi) if xi < 0. Show that ????̂

n(Zn) = ????̂

n(Z∗

n ).