Optimality

136 D. M. Dabrowska
(ii) If τ0 is a discontinuity point of the survival function EPY (t), then τ0 =
sup{t : P( ˜ T ≥ t) > 0} < sup{t : P(T ≥ t) > 0}. If τ0 is a continuity
point of this survival function, then τ0 = sup{t : P(T ≥ t) > 0} ≤
sup{t : P( ˜ T≥ t) > 0}.
For P∈P, let A(t) = AP(t) be given by
(2.1) A(t) =
� t
0
ENP(du)
EPY (u) .
If the censoring time ˜ T is independent of covariates, then A(t) reduces to the
marginal cumulative hazard function of the failure time T, restricted to the interval
[0, τ0]. Under Assumption 2.2 this parameter forms in general a function of
the marginal distribution of covariates, and conditional distributions of both failure
and censoring times. Nevertheless, we shall find it, and the associated Aalen–Nelson
estimator, quite useful in the sequel. In particular, under Assumption 2.2, the conditional
cumulative hazard function H(t|z) of T given Z is uniformly dominated by
A(t). We have
and
A(t) =
� t
0
H(dt|z)
A(dt) =
E[α(Γ0(u−), θ0, Z)|X≥ u]Γ0(du)
α(Γ0(t−), θ0, z)
Eα(Γ0(t−), θ0, Z)|X≥ t) ,
for t≤τ(z) = sup{t : EY (t)|Z = z > 0} and µ a.e. z. These identities suggest to
define a parameter ΓP,θ as solution to the nonlinear Volterra equation
(2.2)
ΓP,θ(t) =
=
� t
0
� t
0
EPN(du)
EPY (u)α(Γθ(u−), θ, Z)
AP(du)
EPα(Γθ(u−), θ, Z)|X≥ u) ,
with boundary condition ΓP,θ(0−) = 0. Because Conditions 2.2 are not needed to
solve this equation, we shall view Γ as a map of the setP× Θ intoX =∪{X(P) :
P∈P}, where
X(P) ={g : g increasing, e −g ∈ D(T ), g≪ EPN, m −1
2 AP≤ g≤ m −1
1 AP}
and m1, m2 are constants of Condition 2.1(iii). Here D(T ) denotes the space of
right-continuous functions with left-hand limits, and we chooseT = [0, τ0], if τ0
is a discontinuity point of the survival function EPY (t), andT = [0, τ0), if it is a
continuity point. The assumption g≪EPN means that the functions g inX(P)
are absolutely continuous with respect to the sub-distribution function EPN(t).
The monotonicity condition implies that they admit integral representation g(t) =
� t
0 h(u)dEPN(u) and h≥0, EPN-almost everywhere.
2.2. Estimation of the transformation
Let (Ni, Yi, Zi), i = 1, . . . , n be an iid sample of the (N, Y, Z) processes. Set
S(x, θ, t) = n −1 � n
i=1 Yi(t)α(x, θ, Zi) and denote by ˙ S, S ′ the derivatives of these