Optimality

138 D. M. Dabrowska
2.3. Some auxiliary notation
From now on we assume that the function EN(t) is continuous. We shall need some
auxiliary notation. Define
e[f](u, θ) = E{Y (u)[fα](Γθ(u), θ, Z)}
E{Yi(u)α(Γθ(u), θ, Z)} ,
where f(x, θ, Z), is a function of covariates. Likewise, for any two such functions, f1
and f2, let cov[f1, f2](u, θ) = e[f1f T 2 ](u, θ)−(e[f1]e[f2] T )(u, θ) and var[f](u, θ) =
cov[f, f](u, θ). We shall write
e(u, θ) = e[ℓ ′ ](u, θ), ē(u, θ) = e[ ˙ ℓ](u, θ),
v(u, θ) = var[ℓ ′ ](u, θ), ¯v(u, θ) = var[ ˙ ℓ](u, θ), ρ(u, θ) = cov[ ˙ ℓ, ℓ ′ ](u, θ),
for short. Further, let
(2.4)
Kθ(t, t ′ ) =
Bθ(t) =
and define
� �
(2.5) κθ(τ) =
� t∧t ′
0 � t
0
0