Optimality

Semiparametric transformation models 161
We decompose the process Un(θ) as Un(θ) = Ûn(θ) + Ūn(θ), where
Ûn(θ) = 1
n
Ūn(θ) =− 1
n
n�
i=1
� τ
n�
i=1
0
� τ
We have Ûn(θ) = � 3
j=1 Unj(θ), where
[bi(Γnθ(t), θ, t)−b2i(Γnθ(t), θ, t)ϕθ0(t)]Ni(dt),
0
b2i(Γnθ(t), θ, t)[ϕnθ− ϕθ0](t)]Ni(dt).
Un1(θ) = Ũn1(θ) + Bn1(τ, θ)−
Un2(θ) =
Un3(θ) =
� τ
0
� τ
0
� τ
[Γnθ− Γθ](t)ˆr1(dt, θ),
[Γnθ− Γθ](t)ˆr2(dt, θ).
0
ϕθ0(u)Bn2(du, θ),
As in Section 2.4, b1i(x, θ, t) = ˙ ℓ(x, θ, Zi)−[ ˙ S/S](x, θ, t) and b2i(x, θ, t) = ℓ ′ (x, θ, Zi)
−[S ′ /S](x, θ, t). If ˙ bpi and b ′ pi are the derivatives of these functions with respect to
θ and x, then
ˆr1(s, θ) = 1
n
ˆr2(s, θ) = 1
n
n�
� s
i=1 0
� s � 1
n�
i=1
0
[b ′ 1i(Γθ(t), θ, t)−b ′ 2i(Γθ(t), θ, t)ϕθ0(t)]Ni(dt),
0
ˆr2i(t, θ, λ)dλNi(dt),
ˆr2i(t, θ, λ) = [b ′ 1i(Γθ(t) + λ(Γnθ− Γθ)(t), θ, t)−b ′ 1i(Γθ(t), θ, t)]
− [b ′ 2i(Γθ(t) + λ(Γnθ− Γθ)(t), θ,t)−b ′ 2i(Γθ(t), θ, t)] ϕθ0(t).
We also have Ūn(θ) = Un4(θ) + Un5(θ), where
Un4(θ) =−
Un5(θ) = 1
n
Bn(t, θ) = 1
n
= 1
n
� τ
0
n�
[ϕnθ− ϕθ0](t)Bn(dt, θ),
� τ
i=1 0
� t
n�
i=1 0
� t
n�
i=1
0
[ϕnθ−ϕθ0](t)[b2i(Γnθ(u), θ, u)−b2i(Γθ(u), θ, u)]Ni(dt),
b2i(Γθ(u), θ, u)Ni(du)
˜ b2i(Γθ(u), θ, u)Mi(du, θ) + B2n(t, θ).
We first show that Ūn(θ0) = oP(n −1/2 ). By Lemma 5.1, √ nB2n(t, θ0) converges
in probability to 0, uniformly in t. At θ = θ0, the first term multiplied by √ n converges
weakly to a mean zero Gaussian martingale. We have�ϕnθ0−ϕθ0�∞ = oP(1),
�ϕθ0�v