Consider the estimator ????̂1 defined by the one-step Newton–Raphson procedure defined in Section 2.10.5.1. Assume that the underlying distribution is symmetric about ????, that ???? is odd and differentiable, and that the initial estimator ????̂0 is consistent for ????.

(a) Show that ????̂1 is consistent for ????.

(b) If ???? is twice differentiable, show that ????̂1 has the same influence function as the M-estimator ????̂ defined by ave{????(x − ????)} = 0 (and hence, by (3.18),

????̂1 has the same asymptotic variance as ????̂).

(c) If ???? is bounded and ????′

(x) > 0 for all x, and the asymptotic BP of ????̂0 is 0.5, show that also ????̂1 has an asymptotic BP of 0.5.