Optimality

Semiparametric transformation models 155
m2[m1EPY (τ)] −1 ,|gθ,t(Z)−gθ ′ ,t(Z)|≤|θ−θ ′ |h1(τ),|gθ,t(Z)−gθ,t ′(Z)|≤[P(X∈
[t∧t ′ , t∨t ′ )) + P(X∈ (t∧t ′ , t∨t ′ ], δ = 1)]h2(τ), where
h1(τ) = 2m2[m1EPY (τ)] −1 [ψ1(d0AP(τ)) + ψ(0)d1 exp[d2AP(τ)],
h2(τ) = m2[m1EPY (τ)] −2 [m2 + 2ψ(0)].
Setting h(τ) = max[h1(τ), h2(τ), m2(m1EPY (τ)) −1 ], it is easy to verify that the
class of functions{fθ,t(x, δ, z)/h(τ) : θ∈Θ, t≤τ} is Euclidean for a bounded envelope.
The law of iterated logarithm for empirical processes over Euclidean classes of
functions [1] implies therefore that part (iii) is satisfied by the process V = S/s−1.
For the remaining choices of the V processes the proof is analogous and follows
from the Glivenko–Cantelli theorem for Euclidean classes of functions [29].
6. Proof of Proposition 2.1
6.1. Part (i)
For P∈P, let A(t) = AP(t) be given by (2.1) and let τ0 = sup{t : EPY (t) > 0}.
The condition 2.1 (ii) assumes that there exist constants m1 < m2 such that the
hazard rate α(x, θ|z) is bounded from below by m1 and from above by m2. Put A1 =
m −1
1 A(t) and A2(t) = m −1
2 (t). Then A2≤ A1. Further, Condition 2(iii) assumes
that the function ℓ(x, θ, z) = log α(x, θ, z) has a derivative ℓ ′ (x, θ, z) with respect to
x satisfying|ℓ ′ (x, θ, z)|≤ψ(x) for some bounded decreasing function. Suppose that
ψ≤cand define ρ(t) = max(c,1)A1(t). Finally, the derivative ˙ ℓ(x, θ, z) satisfies
| ˙ ℓ(x, θ, z)| ≤ ψ1(x) for some bounded function or a function that is continuous
strictly increasing, bounded at origin and satisfying � ∞
0 ψ1(x) 2e−xdx