Consider the CUBIF estimator defined in Section 7.3.

(a) Show that he correction term c

(a,

b) defined above (7.26) is a solution of the equation Em−1(a)(????H b (y − g

(a) − c

(a, b))) = 0.

(b) In the case of the logistic model for the Bernoulli family put g

(a) = ea∕

(1 + ea). Then prove that c

(a,

b) = c∗(g(a), b), where c∗(p,

b) =

⎧

⎪

⎪

⎨

⎪

⎪

⎩

(1 − p)(p − b)∕p if p > max (1 2 , b

)

p(b − 1 + p)∕(1 − p) if p < min (1 2 , b

)

0 elsewhere.

(c) Show that the limit when b → 0 of the CUBIF estimator for the model in Problem 7.3 satisfies the equation

∑n i=1

(y − p(xi, ????))

max(p(xi, ????), 1 − p(xi, ????)) sgn(xi) = 0.

Compare this estimator with the one of Problem 7.5.

(d) Show that the influence function of this estimator is IF(y, x, ????) = 1 A

(y − p(x, ????))sgn(xi)

max(p(x, ????), 1 − p(x, ????))

with A = E(min(p(x, ????)(1 − p(x, ????))|x|); and that the gross error sensitivity is GES(????) = 1∕A.

(e) Show that this GES is smaller than the GES of the estimator given in Problem