Optimality

Semiparametric transformation models 151
n −1 � n
i=1 1(Xi ≥ t) and Further, let �·� be the supremum norm in the set
ℓ ∞ ([0, τ]×Θ), and let�·�∞ be the supremum norm ℓ ∞ ([0, τ]). We assume that
the point τ satisfies Condition 2.0(iii).
Define
Rn(t, θ) =
Rpn(t, θ) =
Rpn(t, θ) =
R9n(t, θ) =
R10n(t, θ) =
Rpn(t, θ) =
Bpn(t, θ) =
� t
0
� t
0
� t
0
N.(du)
S(Γθ(u−), θ, u) −
� t
EN(du)
0 s(Γθ(u−), θ, u) ,
h(Γθ(u−), θ, u)[N.− EN](du), p = 5,6,
Hn(Γθ(u−), θ, u)N(du)
� t
− h(Γθ(u−), θ, u)EN(du), p = 7, 8,
� 0 �
EN(du)| Pθ(u, w)R5n(dw, θ)|,
[0,t)
� t
0
� t
0
� t
0
(u,t]
√ nRn(u−, θ)R5n(du, θ),
Hn(Γnθ(u−), θ, u)N.(du)
−
� t
0
h(Γθ(u−), θ, u)EN(du), p = 11,12,
Fpn(u, θ)Rn(du, θ), p = 1, 2,
wherePθ(u, w) is given by (2.3). In addition, Hn = K ′ n for p = 7 or p = 11,
Hn = ˙ Kn for p = 8 or p = 12, h = k ′ for p = 5, 7 or p = 11, and h = ˙ k for p = 6, 8
or 12. Here k ′ =−[s ′ /s 2 ], ˙ k =−[˙s/s 2 ], K ′ n =−[S ′ /S 2 ], ˙ Kn =−[ ˙ S/S 2 ]. Further,
set F1n(u, θ) = [ ˙ S− ēS](Γθ(u), θ, u) and F2n(u, θ) = [S ′ − eS ′ ](Γθ(u), θ, u).
Lemma 5.1. Suppose that Conditions 2.0 and 2.1 are satisfied.
(i) √ nRn(t, θ) converges weakly in ℓ ∞ ([0, τ]×Θ) to a mean zero Gaussian process
R whose covariance function is given below.
(ii)�Rpn�→0 a.s., for p = 5, . . . ,12.
(iii) √ n�Bpn�→0 a.s. for p = 1, 2.
(iv) The processes Vn(Γθ(t−), θ, t) and Vn(Γθ(t), θ, t), where Vn = S/s−1 satisfy
�Vn� = O(bn) a.s. In addition,�Vn�→0a.s. for Vn = [S ′ − s ′ ]/s,[S ′′ −
s ′′ ]/s,[ ˙ S− ˙s]/s,[ ¨ S−¨s]/s and [ ˙ S ′ − ˙s ′ ]/s.
Proof. The Volterra identity (2.2), which defines Γθ as a parameter dependent on
P, is used in the foregoing to compute the asymptotic covariance function of the
process R1n. In Section 6 we show that the solution to the identity (2.2) is unique
and, for some positive constants d0, d1, d2, we have
(5.1)
Γθ(t)≤d0AP(t), |Γθ(t)−Γθ ′(t)|≤|θ− θ′ |d1 exp[d2AP(t)],
|Γθ(t)−Γθ(t ′ )| ≤ d0|AP(t)−AP(t ′ )|
≤
d0
EPY (τ) P(X∈ (t∧t′ , t∨t ′ ], δ = 1),
with similar inequalities holding for the left continuous version of Γθ = Γθ,P. Here
AP(t) is the cumulative hazard function corresponding to observations (X, δ).