On the Identification of Misspecified Propensity Scores - School of ...

Thus, the second condition of the equicontinuity lemma is also satisfied.
Next, we verify the same conditions for the process given in (14). First, we observe that
we can rewrite each term of that summation as
n √ hnεj0
E
an(n − 1)hn
an[Q(Xi, θ) − Q(Xi, θ0)]K
Q(Xi, θ) − Q(Xj, θ)
|Xj .
Since Q is Lipschitz with Lipschitz constant LQ, each of these terms is bounded by
n √ hn|εj0|LQǫθ
an(n − 1) E
1
hn
K
Q(Xi, θ) − Q(Xj, θ)
|Xj .
hn
Then using the Lipschitz continuity of K and Q, we have
n √ hn|εj0|LQǫθ
an(n − 1) E
1
hn
K
Q(Xi, θ) − Q(Xj, θ)
|Xj
hn
= n|εj0|LQǫθ
an(n − 1) √
Q(Xi, θ) − Q(Xj, θ) Q(Xi, θ0) − Q(Xj, θ0)
E K
−K
|Xj +
hn
hn
hn
n|εj0|LQǫθ
an(n − 1) √
Q(Xi, θ0) − Q(Xj, θ0)
E K
|Xj
hn
hn
nC
≤
a2 nh 3/2
n (n − 1) |εj0| + n√hn|εj0|LQǫθ an(n − 1) E
1
hn
K
Q(Xi, θ0) − Q(Xj, θ0)
|Xj ,
hn
where C = 2LKL2 gǫ2 θ . Then using a change of variables operation, we see that the second term
in the last expression equals
n √ hn|εj0|LQǫθ
an(n − 1)
hn
|K (u)| fQ Q(Xj, θ0) + uhn du.
Since K is bounded the integrals in this last expression are all finite. Therefore,
n √ hn
an(n − 1) |εj0|
1
,
anh2 C + C1
n
where C1 is the appropriate constant, is an envelope for (14). To verify the first condition of
the equicontinuity lemma, consider
n
lim sup
n→∞
j
2hn a2 n(n − 1) 2 E ε 2
j0
1
anh2 2 nhn
C + C1 ≤ lim sup
n
n→∞ a2
1
n anh2 2 C + C1 .
n
Sufficient conditions for this to be finite are that limn→∞ nhna −2
n < ∞, and limn→∞ 1
anh 2 n
< ∞,
which are implied by our assumptions. Thus the first condition of the equicontinuity lemma
is satisfied.
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