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- ∂gn(X, θ̇) ∂θ′ # t, THE EXPONENTIALLY TILTED HELLINGER DISTANCE ESTIMATOR 41 with θ̇ ∈ (θ∗ , θn) and may vary from row to row. Noting that EPθn,ζn [g(X, θn)] = 0, EPθn,ζn [gn(X, θn)] = EPθn,ζn [g(X, θn)I{X / ∈ Xn}] = o(n−1/2 ) (we refer to Equation A.16 of KOE (2013b) for the proof). Also, thanks to Assumption 3(iv), and by the continuity of the map θ 7→ EP∗ h ∂g(X,θ) ∂θ′ i in a neighborhood of θ∗ , we can claim that EPθn,ζn h ∂gn(X,θ̇) ∂θ′ i
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- From Lemma A.4 of KOE (2013b), we have EP∗ (gn(X, θ∗ )) = o(n−1/2 ). Thus, − √ nF Z gn(X, θ∗ )dQn + o(1) = − √ nF Z gn(X, θ∗ )(dQn − dP∗) + o(1) = − √ nF Z gn(X, θ∗ ) dQ1/2 n − dP 1/2 ∗ dQ1/2 n − √ nF Z gn(X, θ∗ ) dQ1/2 n − dP 1/2 ∗ dP 1/2 ∗ + o(1). By the triangle inequality, we have n( Ä ◦ T̄(Qn) − Ä(θ∗ ) 2 ≤ n(A1 + A2 + 2A3) + o(1), with A1 = F Z gn(x, θ∗ ) dQ1/2 n − dP 1/2 ∗ dQ1/2 n 2 , A1 = F Z gn(x, θ∗ ) dQ1/2 n − dP 1/2 ∗ dP 1/2 ∗ 2
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- Proof of Theorem 4.1: The proof follows similar lines as those of Theorem 3.1(ii) in KOE (2013a). To establish Fisher consistency, let Pθ,ζ be a regular sub-model such that for t ∈ Rp , Pθn,ζn ∈ BH(P∗, r/ √ n) for n large enough, with θn = θ∗ + t/ √ n and ζn = O(n−1/2 ). We further assume that EPθn,ζn [supθ∈Θ kg(X, θ)kα ] ≤ δ < ∞ for some δ > 0. (Note that the particular sub-model used by KOE to derive the lower bound in their Theorem 3.1(i) satisfies this condition.) We have to show that √ n(T̄(Pθn,ζn ) − θ∗ ) → t, as n → ∞. From Lemma C.5, √ n(T̄(Pθn,ζn ) − θ∗ ) = −ΣG′ Ω−1 √ nEPθn,ζn [gn(X, θ∗ )] + o(1). By a mean-value expansion, we have: √ nEPθn,ζn [gn(X, θ∗ )] = √ nEPθn,ζn [gn(X, θn)] − EPθn,ζn
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