Optimality

158 D. M. Dabrowska
Next suppose that τ0 is a continuity point of this survival function, and let
T = [0, τ0). We have sup t∈T|exp[−Apn(t)−exp[−Ap(t)| = oP(1). In addition, for
any τ < τ0, we have exp[−A1n(τ)] ≤ exp[−Γnθ(τ)] ≤ exp[−A2n(τ)]. Standard
monotonicity arguments imply sup t∈T|exp[−Γnθ(t)−exp[−Γθ(t)| = oP(1), because
Γθ(τ)↑∞ as τ↑∞.
6.2. Part (iii)
The process ˆ W(t, θ) = √ n[Γnθ− Γθ](t) satisfies
where
Define
ˆW(t, θ) = √ �
nRn(t, θ)−
b ∗ nθ(u) =
�� 1
0
[0,t]
ˆW(u−, θ)N.(du)b ∗ nθ(u),
� ′ 2
S /S � �
(θ, Γθ(u−) + λ[Γnθ− Γθ](u−), u)dλ .
˜W(t, θ) = √ nRn(t, θ)−
� t
where bθ(u) = [s ′ /s 2 ](Γθ(u), θ, u). We have
and
where
˜W(t, θ) = √ nRn(t, θ)−
ˆW(t, θ)− ˜ W(t, θ) =−
�
rem(t, θ) =−
[0,t]
� t
The remainder term is bounded by
� τ
0
0
� t
0
0
˜W(u−, θ)bθ(u)EN(du),
√ nRn(u−, θ)bθ(u)EN(du)Pθ(u, t)
[ ˆ W− ˜ W](u−, θ)b ∗ nθ(u)N.(du) + rem(t, θ),
˜W(u−, θ)[b ∗ nθ(u)N.(du)−bθ(u)EN(du)].
| ˜ W(u−, θ)||[b ∗ nθ− bθ](u)|N.(du) + R10n(t, θ)
+
� t−
0
| √ nRn(u−, θ)||bθ(u)|R9n(du, θ).
By noting that R9n(·, θ) is a nonnegative increasing process, we have�rem� =
oP(1) +�R10n� + OP(1)�R9n� = oP(1). Finally,
| ˆ W(t, θ)− ˜ W(t, θ)|≤|rem(t, θ)| +
� t
0
| ˆ W− ˜ W|(u−, θ)ρn(du).
By Gronwall’s inequality (Section 9), we have ˆ W(t, θ) = ˜ W(t, θ) + oP(1) uniformly
in t ≤ τ, θ ∈ Θ. This verifies that the process √ n[Γnθ− Γθ] is asymptotically
Gaussian, under the assumption that observations are iid, but Condition 2.2 does
not necessarily hold.