Optimality

2.1. The model
Semiparametric transformation models 135
Throughout the paper we assume that (X, δ, Z) is defined on a complete probability
space (Ω,F, P), and represents a nonnegative withdrawal time (X), a binary indicator
(δ) and a vector of covariates (Z). Set N(t) = 1(X≤ t, δ = 1), Y (t) = 1(X≥ t)
and let τ0 = τ0(P) = sup{t : EPY (t) > 0}. We shall make the following assumption
about the ”true” probability distribution P.
Condition 2.0. P∈P whereP is the class of all probability distributions such
that
(i) The covariate Z has a nondegenerate marginal distribution µ and is bounded:
µ(|Z|≤C) = 1 for some constant C.
(ii) The function EPY (t) has at most a finite number of discontinuity points, and
EPN(t) is either continuous or discrete.
(iii) The point τ > 0 satisfies inf{t : EP[N(t)|Z = z] > 0} < τ for µ a.e. z. In
addition, τ = τ0, if τ0 is an discontinuity point of EPY (t), and τ < τ0, if τ0
is a continuity point of EPY (t).
For given τ satisfying Condition 2.0(iii), we denote by�·�∞ the supremum
norm in ℓ ∞ ([0, τ]). The second set of conditions refers to the core model{A(·, θ|z) :
θ∈Θ}.
Condition 2.1. (i) The parameter set Θ⊂R d is open, and θ is identifiable in
the core model: θ�= θ ′ iff A(·, θ|z)�≡ A(·, θ ′ |z) µ a.e. z.
(ii) For µ almost all z, the function A(·, θ|z) has a hazard rate α(·, θ|z). There
exist constants 0 < m1 < m2