Optimality

Semiparametric transformation models 141
the case of generalized proportional hazards intensity models. Using results of section
6, we can show that the sample analogue of the function ˙ Γθ satisfies Conditions
2.3(iv) and 2.3(v).
Example 2.3. The logarithmic derivatives of ℓ(x, θ, Z) = log α(x, θ, Z) may be
difficult to compute in some models, so we can try to replace them by different
functions. In particular, suppose that h(x, θ, Z) is a differentiable function with
respect to both arguments and the derivatives satisfy a similar Lipschitz continuity
assumption as in condition 2.1. Consider the score process (2.6) with function
ϕθ = 0 and weights b1i(x, t, θ) = h(x, θ, Zi)−[Sh/S](x, θ, t) where Sh(x, θ, t) =
� n
i=1 Yi(u)[hα](x, θ, Zi), and ϕnθ≡ 0. For p = 0 and p = 2, define matrices Σ h pϕ by
replacing the functions vϕ and ρϕ appearing in matrices Σ0ϕ and Σ2ϕ with
v h ϕ(t, θ0) = var[h(Γθ0(X), θ0, Z)|X = t, δ = 1],
ρ h ϕ(t, θ0) = cov[h(Γθ0(Xi), θ0, Zi), ℓ ′ (Γθ0(Xi), θ0, Zi)|X = t, δ = 1].
The matrix Σ1ϕ(θ0, τ) is changed to Σ h 1ϕ(θ0, τ) = � ¯ρ h ϕ(t, θ0)EN(du), where the
integrand is equal to
cov[h(Γθ0(X), θ0, Z), ˙ ℓ(Γθ0(X), θ0, Z) + ℓ ′ (Γθ0(X), θ0, Z) ˙ Γθ0(X)|X = t, δ = 1].
The statement of Proposition 2.2 remains valid with matrices Σpϕ replaced by
Σ h pϕ, p = 1, 2, provided in Condition 2.3 we assume that the matrix Σ h 0ϕ is positive
definite and the matrix Σ h 1ϕ is non-singular. The resulting estimates have a structure
analogous to that of the M-estimates considered in the case of uncensored data by
Bickel et al. [6] and Cuzick [13]. Alternatively, instead of functions ˙ ℓi(x, θ, z) and
ℓ ′ (x, θ, z), the weight functions b1i and b2i can use logarithmic derivatives of a
different distribution, with the same parameter θ. The asymptotic variance is of
similar form as above. In both cases, the derivations are similar to Section 7, so we
do not consider analysis of these score processes in any detail.
Example 2.4. Our final example shows that we can choose the ϕθ function so that
the asymptotic variance of the estimate ˆ θ is equal to the inverse of the asymptotic
variance of the normalized score process, √ nUn(θ0). Remark 2.1 implies that if
ρ−Γ ˙ (u, θ0)≡0 but v(u, θ0)�≡ 0 a.e. EN, then for ϕθ =− ˙ Γθ the matrices Σq,ϕ, q =
1,2 are equal. This also holds for v(u, θ0)≡0. We shall consider now the case
v(u, θ0)�≡ 0 and ρ−Γ ˙ (u, θ0)�≡ 0 a.e. EN, and without loss of generality, we shall
assume that the parameter θ is one dimensional.
We shall show below that the equation
(2.7)
ϕθ(t) +
� τ
0
=− ˙ Γθ(t) +
Kθ(t, u)v(u, θ)ϕθ(u)EN(du)
� τ
0
Kθ(t, u)ρ(u, θ)EN(du)
has a unique solution ϕθ square integrable with respect to the measure (2.4). For θ =
θ0, the corresponding matrices Σ1,ϕ(θ0, τ) and Σ2,ϕ(θ0, τ) are finite. Substitution
of the conditional correlation function ρϕ(t, θ0) = ρ(t, θ0)−ϕθ0(t)v(t, θ0) into the
matrix Σ2,ϕ(θ0, τ) shows that they are also equal. (In the multiparameter case, the
equation (2.7) is solved for each component of the θ).
Equation (2.7) simplifies if we replace the function ϕθ by ψθ = ϕθ + ˙ Γθ. We get
� τ
(2.8) ψθ(t)−λ
0
Kθ(t, u)ψθ(u)Bθ(du) = ηθ(t),