Optimality

152 D. M. Dabrowska
To show part (i), we use the quadratic expansion, similar to the expansion of the
ordinary Aalen–Nelson estimator in [19]. We have Rn = �4 j=1 Rjn,
� t �
R1n(t, θ) = 1
n
= 1
n
R2n(t, θ) = −1
n 2
R3n(t, θ) = −1
n 2
R4n(t, θ) =
� t
0
n�
i=1
n�
i=1
�
0
Ni(du) Si
−
s(Γθ(u−), θ, u) s2 (Γθ(u−),
�
θ, u)EN(du)
R (i)
1n (t, θ),
� t
i�=j
0
� t
n�
i=1
0
� S− s
s
� Si− s
s 2
� Si− s
� 2
s 2
�
(Γθ(u−), θ, u)[Nj− ENj](du),
�
(Γθ(u−), θ, u)[Ni− ENi](du),
N.(du)
(Γθ(u−), θ, u)
S(Γθ(u−), θ, u) ,
where Si(Γθ(u−), θ, u) = Yi(u)α(Γθ(u−), θ, Zi).
The term R3n has expectation of order O(n−1 ). Using Conditions 2.1, it is easy
to verify that R2n and n[R3n− ER3n] form canonical U-processes of degree 2
and 1 over Euclidean classes of functions with square integrable envelopes. We
have�R2n� = O(b2 n) and n�R3n− ER3n� = O(bn) almost surely, by the law of
iterated logarithm for canonical U processes [1]. The term R4n can be bounded by
�R4n�≤�[S/s]−1� 2m −1
1 An(τ). But for a point τ satisfying Condition 2.0(iii), we
have An(τ) = A(τ)+O(bn) a.s. Therefore part (iv) below implies that √ n�R4n�→0
a.s.
The term R1n decomposes into the sum R1n = R1n;1− R1n;2, where
� t
R1n;1(t, θ) = 1
n
R1n;2(t, θ) =
n�
i=1
� t
0
0
Ni(du)−Yi(u)A(du)
,
s(Γθ(u−), θ, u)
G(u, θ)Cθ(du)
and G(t, θ) = [S(Γθ(u−), θ, u)−s(Γθ(u−), θ, u)Y.(u)/EY (u)]. The Volterra identity
(2.2) implies
ncov(R1n;1(t, θ), R1n;1(t ′ , θ ′ )) =
ncov(R1n;1(t, θ), R1n;2(t ′ , θ ′ ))
=
� t � ′
u∧t
0 0
� t � ′
u∧t
−
0
0
ncov(R1n;2(t, θ), R1n;2(t ′ , θ ′ ))
=
� t � ′
t ∧u
0 0
� ′
t � t∧v
+
−
0 0
� ′
t∧t
0
� t∧t ′
0
[1−A(∆u)]Γθ(du)
s(Γθ ′(u−), θ′ , u) ,
E[α(Γθ ′(v−), Z, θ′ |X = u, δ = 1]Cθ ′(dv)Γθ(du)
Eα(Γθ ′(v−), Z, θ′ |X≥ u]]Cθ ′(dv)Γθ(du),
f(u, v, θ, θ ′ )Cθ(du)Cθ ′(dv)
f(v, u, θ ′ , θ)Cθ(du)Cθ ′(dv)
f(u, u, θ, θ ′ )Cθ ′(∆u)Cθ(du),