Optimality

Semiparametric transformation models 153
where f(u, v, θ, θ ′ ) = EY (u)cov(α(Γθ(u−), θ, Z), α(Γθ ′(v−), θ′ , Z)|X ≥ u). Using
CLT and Cramer-Wold device, the finite dimensional distributions of √ nR1n(t, θ)
converge in distribution to finite dimensional distributions of a Gaussian process.
The process R1n can be represented as R1n(t, θ) = [Pn− P]ht,θ, where H =
{ht,θ(x, d, z) : t≤τ, θ∈Θ} is a class of functions such that each ht,θ is a linear
combination of 4 functions having a square integrable envelope and such that
each is monotone with respect to t and Lipschitz continuous with respect to θ. This
is a Euclidean class of functions [29] and{ √ nR1n(t, θ) : θ∈Θ, t≤τ} converges
weakly in ℓ ∞ ([0, τ]×Θ) to a tight Gaussian process. The process √ nR1n(t, θ) is
asymptotically equicontinuous with respect to the variance semimetric ρ. The function
ρ is continuous, except for discontinuity hyperplanes corresponding to a finite
number of discontinuity points of EN. By the law of iterated logarithm [1], we also
have�R1n� = O(bn) a.s.
Remark 5.1. Under Condition 2.2, we have the identity
ncov(R1n;2(t, θ0), R1n;2(t ′ , θ0))
2�
= ncov(R1n;p(t, θ0, R1n;3−p(t ′ , θ0))
p=1
�
−
[0,t∧t ′ ]
EY (u)var(α(Γθ0(u−)|X≥ u)Cθ0(∆u)Cθ0(du).
Here θ0 is the true parameter of the transformation model. Therefore, using the assumption
of continuity of the EN function and adding up all terms,
ncov(R1n(t, θ0), R1n(t ′ , θ0)) = ncov(R1n;1(t, θ0), R1n;1(t ′ , θ0)) = Cθ0(t∧t ′ ).
Next set bθ(u) = h(Γθ(u−), θ, u), h = k ′ or h = ˙ h. Then � t
0 bθ(u)N.(du) = Pnft,θ,
where ft,θ = 1(X ≤ t, δ = 1)h(Γθ(X∧ τ−), θ, X∧ τ−). The conditions 2.1 and
the inequalities (5.1) imply that the class of functions{ft,θ : t ≤ τ, θ ∈ Θ} is
Euclidean for a bounded envelope, for it forms a product of a VC-subgraph class
and a class of Lipschitz continuous functions with a bounded envelope. The almost
sure convergence of the terms Rpn, p = 5,6 follows from Glivenko–Cantelli theorem
[29].
Next, set bθ(u) = k ′ (Γθ(u−), θ, u) for short. Using Fubini theorem and
|Pθ(u, w)|≤exp[ � w
u |bθ(s)|EN(ds)], we obtain
R9n(t, θ) ≤
�
(0,t)
�
+
EN(du)|R5n(t, θ)−R5n(u, θ)|
(0,t)
�
≤ 2�R5n�
≤ 2�R5n�
�
EN(du)|
[0,t)
� τ
uniformly in t≤τ, θ∈Θ.
0
(u,t]
�
EN(du)[1 +
�
EN(du)exp[
Pθ(u, s−)bθ(s)EN(ds)[R5n(t, θ)−R5n(s, θ)]|
(u,t]
(u,τ]
|P(u, w−)||bθ(w)|EN(dw)]
|bθ|(s)EN(ds)]→0 a.s.