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- | ï¬ âˆˆ L & ï² ï€¾ 0} (argued in the proof of Lemma 1). By applying the functional CLT for bootstrap empirical processes (e.g., Theorem 3.6.1 of VW, 1996; or Theorem 2.6 of Kosorok, 2008), each of two parts within the curly braces on the RHS of (85), normalized by (1 √ ï®), conditionally weakly converges to some tight random function in probability. Therefore, we have the LHS of (85) ï =⇒ # 0 uniformly over (ï¬ï€» ï²) ∈ L × {ï²1     ï²ïŠ }  where we also have used the boundedness of 1∗ (ï¬ï€» ï²) (Assumption 2), the convergence property of ï€¢ï® (ï¬ï€» ï²) and ï‚²ï® (ï¬ï€» ï²), and the fact that the (usual) convergence in probability implies the conditional weak convergence in probability. This completes the proof of Lemma 7.
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- A Appendix This appendix is divided into two subsections — the first subsection dealing with Theorem 1 and the second one with Theorem 2. In this Appendix, terminologies related to empirical process theory follow those of van der Vaart and Wellner (1996), which is hereafter referred to as VW (1996). A.1 Lemmas and proofs for Theorem 1 Step 1: Linearization of the nonparametric estimators ˆ s (ï¬) and ˆ sï‚° (ï¬).
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- G∗  (72) Now, we have obtained the weak convergence of √ ï®[ˆ Ïï‚° − Ï∗ ï‚°] ((ii) of Lemma 4), the conditional weak convergence of √ ï®[ˆ Ï# ï‚° −ˆ Ïï‚°] (the result (72)), and the Hadamard differentiability of Ψ (Lemma 5). By these three results, we can apply the functional delta method for bootstrap (Theorem 3.9.11 of VW, 1996), which verifies that √ ï®( ˆ ï•# (ï‚°) − ˆ ï• (ï‚°)) ï =⇒ #
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- Note that ˆ G# ï‚° is simply in the form of a bootstrapped empirical process, to which we can directly apply the bootstrap CLT (e.g., Theorem 3.6.1 of VW, 1996; or Theorem 2.6 of Kosorok, 2008). That is, by the weak-convergence/Donsker result of ˆ Gï‚° ((i) of Lemma 4), it holds that ˆ G# ï‚° ï =⇒ # G∗  where the conditional weak convergence in probability take places in the space ï¬âˆž(X × L) (given the original observations {ïšï©}ï® ï©=1). This, together with (68) and (70), implies that √ ï®[ˆ Ï# ï‚° − ˆ Ïï‚°] ï =⇒ #
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- Proof. Recall the functional form of Φ, the linear specification in (7), and the form of the likelihood function (8). Then, we can easily verify the existence of the Hadamard derivative and its form (62) by using the standard chain rule for Hadamard differentiable functionals and arguments analogous to those in the proof of Lemma 20.10 of van der Vaart (1998). We omit details of the proof for brevity. Given these preparations, we can now prove Theorem 1.
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- Proof. We can check the (uniform) validity of the linear representation in (15) by the results proved in Lemmas 1, 2 and 3. Therefore, given the part (i) of the lemma, the result (ii) follows immediately by using standard results on stochastic convergence (see, e.g., (iv) of Theorem 18.10 in van der Vaart, 1998) and we omit details. We also omit the proof of the part (iii) for brevity, which is quite analogous to the proof of the parts (i) and (ii).
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