Suppose X1,..., Xn are i.i.d. according to a quadratic mean differentiable model {P, }, where

Suppose X1,..., Xn are i.i.d. according to a quadratic mean differentiable model {Pθ, θ ∈ }, where is an open subset of the real line. Suppose an estimator sequence θˆ

n is asymptotically linear in the sense that, n1/2

(θˆ

n − θ0) = n−1/2n i=1

ψθ0 (Xi) + oPn

θ0

(1)

where Eθ0 [ψθ0 (Xi)] = 0 and τ 2 = V arθ0 [ψθ0 (Xi)] < ∞.

(i) Find the joint limiting behavior of (n1/2(ˆ

θn − θ0), Zn) under θ0 + hn−1/2, where Zn is the normalized score statistic given by Zn = n−1/2n i=1 η˜(Xi, θ0)
and η˜(·, θ0) is the usual score function.
(ii) Find a simple if and only if condition for ˆ
θn to be regular (at θ0). (Your answer should depend on something about the functions ψθ0 (·) and η˜(·, θ0).)
(iii) Find a simple if and only if condition for the statistic Dn ≡ n1/2 (θˆ
n − θ0) − I −1 (θ0)Zn to be asymptotically ancillary at θ0, in the sense that its limiting distribution under θ0 + hn−1/2 does not depend on h.
(iv) Find a simple if and only if condition for Dn and Zn to be asymptotically independent.
(v) Under the additional assumption of regularity, find a simple if and only if condition for the statistic sequence Dn defined above to tend in probability under θ0 to 0 (and hence θˆ
n is efficient under this condition).