Optimality

Semiparametric transformation models 143
where the kernel k is given by k(t, u) = c(t∧u). Since this is the covariance function
of a time transformed Brownian motion, we obtain a simpler equation. The solution
to this Fredholm equation is
(2.12)
˜ ψ(t) =
� τ
0
˜∆(t, u)˜ρ − ˙ G (u, θ0)EN(du),
where ˜ ∆(t, u) = ˜ ∆(t, u,−1), and ˜ ∆(t, u, λ) is the resolvent corresponding to the
kernel k. More generally, we consider the equation
(2.13)
Its solution is of the form
˜ ψ(t) +
� τ
0
˜ψ(t) = ˜η(t)−
k(t, u) ˜ ψ(u)b(du) = ˜η(t).
� τ
0
˜∆(t, u)b(du)˜η(u).
To give the form of the ˜ ∆ function, note that the constant κθ0(τ) defined in (2.5)
satisfies
κ(τ) = κθ0(τ) =
� τ
0
c(u)b(du).
Proposition 2.3. Suppose that Assumptions 2.0(i) and (ii) are satisfied and
v(u, θ0)�≡ 0, For j = 0,1, 2,3, n≥1 and s < t define interval functions Ψj(s, t) =
� ∞
m=0 Ψjm(s, t) as follows:
Ψ00(s, t) = 1, Ψ20(s, t) = 1,
� �
Ψ0n(s, t) =
Ψ0,n−1(s, u1−)c(du1)b(du2) n≥1,
�
s