Optimality

160 D. M. Dabrowska
We have � t
0 |ψ1n(h, θ, u)|N.(du)≤ρn(t) and � t
0 |ψ2n(h, θ, u)|N.(du)≤h T � t
0 Bn(u)×
� τ
N.(du), for a process Bn with limsupn 0 Bn(u)N.(du) = O(1) a.s. This follows
from condition 2.1 and some elementary algebra. By Gronwall’s inequality,
limsupn supt≤τ|remn(h, θ, t)| = O(|h| 2 ) = o(|h|) a.s. A similar argument shows that
if hn is a nonrandom sequence with hn = O(n −1/2 ), then limsup n sup t≤τ|remn(hn,
θ, t)| = O(n−1 ) a.s. If ˆ hn is a random sequence with | ˆ hn|
limsupn supt≤τ|remn( ˆ hn, θ, t)| = Op(| ˆ hn| 2 ).
6.4. Part (iv)
P
→ 0, then
Next suppose that θ0 is a fixed point in Θ, EN(t) is continuous, and ˆ θ is a √ nconsistent
estimate of θ0. Since EN(t) is a continuous function,{ ˆ W(t, θ) : t≤τ, θ∈
Θ} converges weakly to a process W whose paths can be taken to be continuous
with respect to the supremum norm. Because √ n[ ˆ θ−θ0] is bounded in probability,
we have √ n[Γ n ˆ θ − Γθ0]− √ n[ ˆ θ− θ0] ˙ Γθ0 = ˆ W(·, ˆ θ)+ √ n[Γˆ θ − Γθ0− [ ˆ θ− θ0] ˙ Γθ0] =
ˆW(·, ˆ θ)+OP( √ n| ˆ θ−θ0| 2 )⇒W(·, θ0) by weak convergence of the process{ ˆ W(t, θ) :
t≤τ, θ∈Θ} and [8].
7. Proof of Proposition 2.2
The first part follows from Remark 3.1 and part (iv) of Proposition 2.1. Note that at
the true parameter value θ = θ0, we have √ n[Γnθ0−Γθ0](t) = n1/2 � t
0 R1n(du, θ0)×
Pθ0(u, t) + oP(1), where R1n is defined as in Lemma 5.1,
R1n(t, θ) = 1
n
n�
i=1
� t
and Mi(t, θ) = Ni(t)− � t
0 Yi(u)α(Γθ, θ, Zi)Γθ(du).
We shall consider now the score process. Define
Ũn1(θ) = 1
n
Ũn2(θ) =
n�
i=1
� τ
0
� τ
0
0
Mi(du, θ)
s(Γθ(u−), θ, u) .
˜ bi(Γθ(u), θ)Mi(dt, θ),
�
R1n(du, θ)
(u,τ]
Pθ(u, v−)r(dv, θ).
Here ˜ bi(Γθ(u), θ) = ˜ bi1(Γθ(u), θ) − ˜ bi2(Γθ(u), θ)ϕθ0(t) and ˜ b1i(Γθ(t), θ) =
˙ℓ(Γθ(t), θ, Zi)−[˙s/s](Γθ(t), θ, t), ˜ b2i(Γθ(t), θ) = ℓ ′ (Γθ(t), θ, Zi)−[s ′ /s](Γθ(t), θ, t).
The function r(·, θ) is the limit in probability of the term ˆr1(t, θ) given below. Under
Condition 2.2, it reduces at θ = θ0 to
� t
r(·, θ0) =− ρϕ(u, θ0)EN(du)
0
and ρϕ(u, θ0) is the conditional correlation defined in Section 2.3. The terms
√ nŨ1n(θ0) and √ n Ũ2n(θ0) are uncorrelated sums of iid mean zero variables and
their sum converges weakly to a mean zero normal variable with covariance matrix
Σ2,ϕ(θ0, τ) given in the statement of Proposition 2.2.