Optimality

162 D. M. Dabrowska
�[ˆr1− r](·, θ0)�∞ = oP(1), so that the same integration by parts argument implies
that √ nUn2(θ0) = √ n Ũn2(θ0) + oP(1). Finally, √ nUn1(θ0) = √ n Ũn1(θ0) + oP(1),
by Lemma 5.1 and Fubini theorem.
Suppose now that θ varies over a ball B(θ0, εn) centered at θ0 and having radius
εn, εn ↓ 0, √ nεn → ∞. It is easy to verify that for θ, θ ′ ∈ B(θ0, εn) we have
Un(θ ′ )−Un(θ) = −(θ ′ − θ) T Σ1n(θ0) + (θ ′ − θ) T Rn(θ, θ ′ ), where Rn(θ, θ ′ ) is a
remainder term satisfying sup{|Rn(θ, θ ′ )| : θ, θ ′ ∈ B(θ0, εn)} = oP(1). The matrix
Σ1n(θ) is equal to the sum Σ1n(θ) = Σ11n(θ) + Σ12n(θ),
Σ11n(θ) = 1
n
Σ12n(θ) =− 1
n
n�
i=1
� τ
n�
i=1
0
� τ
[g1ig T 2i](Γnθ(u), θ, u) T Ni(du),
0
[fi− Sf/S](Γnθ(u), θ, u)Ni(du),
where Sf(Γnθ(u), θ, u) = n −1 � n
i=1 Yi(u)[αifi](Γnθ(u), θ, u) and
g1i(θ, Γnθ(u), u) = b1i(Γnθ(u), θ)−b2i(Γnθ(u), θ)ϕθ0(u),
g2i(θ, Γnθ(u), u) = b1i(Γnθ(u), θ) + b2i(Γnθ(u), θ) ˙ Γnθ(u)
fi(θ, Γnθ(u), u) = ¨α
α (Γnθ(u), θ, Zi)− ˙α′
α (Γnθ(u), θ, Zi)ϕθ0(u) T
+ ˙ Γnθ(u)[ ˙α′
α (Γnθ(u), θ, Zi)] T
+ α′′
α (Γnθ(u), θ, Zi) ˙ Γnθ(u)ϕθ0(u) T .
These matrices satisfy Σ11n(θ0) →P Σ1,ϕ(θ0, τ) and Σ12n(θ0) →P 0, and
Σ1,ϕ(θ0, τ) = Σ1(θ0) is defined in the statement of Proposition 2.2. By assumption
this matrix is non-singular. Finally, set hn(θ) = θ + Σ1(θ0) −1 Un(θ). It is easy to
verify that this mapping forms a contraction on the set{θ :|θ−θ0|≤An/(1−an)},
where An =|Σ1(θ0) −1 Un(θ0)| = OP(n −1/2 ) and an = sup{|I− Σ1(θ0) −1 Σ1n(θ0) +
Σ1(θ0) −1 Rn(θ, θ ′ )| : θ, θ ′ ∈ B(θ0, εn)} = oP(1). The argument is similar to Bickel
et al. ([6], p.518), though note that we cannot apply their mean value theorem
arguments.
Next consider Condition 2.3(v.2). In this case we have Ûn(θ ′ )− Ûn(θ) =−(θ ′ −
θ) T Σ1n(θ0) + (θ ′ − θ) T ˆ Rn(θ, θ ′ ), where sup{| ˆ Rn(θ, θ ′ ) : θ, θ ′ ∈ B(θ0, εn)} = oP(1).
In addition, for θ∈Bn(θ0, ε), we have the expansion Ūn(θ) = [ Ūn(θ)− Ūn(θ0)] +
Ūn(θ0) = oP(|θ− θ0| + n −1/2 ). The same argument as above shows that the equation
Ûn(θ) has, with probability tending to 1, a unique root in the ball B(θ0, εn).
But then, we also have Un( ˆ θn) = Ûn( ˆ θn) + Ūn( ˆ θn) = oP(| ˆ θn− θ0| + n −1/2 ) =
oP(OP(n −1/2 ) + n −1/2 ) = op(n −1/2 ).
Part (iv) can be verified analogously, i.e. it amounts to showing that if √ n[ ˆ θ−θ0]
is bounded in probability, then the remainder term ˆ Rn( ˆ θ, θ0) is of order oP(| ˆ θ−θ0|),
and Ūn( ˆ θ) = oP(| ˆ θ− θ0| + n −1/2 ).