Estimators based in adaptively trimming cells in the mixture model

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But this property is only possible if the subsequence {j n } n satisfies (22). Now, let us suppose that 2 or 3 hold. Let {j ′ n} n be a subsequence of {j n } n . Obviously, there exists a subsequence {j ∗ n} n of {j ′ n} n which satisfies that for every i ∈ {1, ..., k}, lim inf λ j∗ n n i > 0 and lim ‖ψ j∗ n n i ‖ = ∞ or lim ψ j∗ n n i = ψ i ∈ Φ, and there also exists i 0 such that lim n ‖ψ j∗ n i0 ‖ = ∞, or lim n δ j∗ n i0 = 0. Without loss of generality we can assume that i 0 = 1. As before, we will denote jn ∗ = n, for n ∈ IN. Let D n := { x ∈ A γn : IP θ ∗ n [1/x] > u/2 } . Then IP n [D n /A γn ] > u/2, and arguing as in (20) and (21), we have that u(1−α)/2 ≤ lim inf n IP n (D n ) and, if x ∈ D n , that f θ ∗ n (x) ≤ k2δ n 1 g ψ n 1 (x)/u. (23) On the other hand, in this case, we have that K 2 := sup n sup i M(ψi n ) < ∞. From here and (23) we have that ] IP n L θ ∗ n /γ n ≤ IP n [I Aγn ∩Dn c log f [ ] θn ∗ + IP n IDn log f θ ∗ n ≤ log + (K 2 ) + log (k2/u) + IP n [ IDn log(δ n 1 g ψ n 1 ) ] , which converges to −∞ for the subsequence we are considering, and the proof ends as in the previous case. • Lemma 6.3 If IP is absolutely continuous and there exists δ > 0 such that θ P exists N 0 ∈ IN such that if n ≥ N 0 , then θ P ∈ Θ n γ n ,u. ∈ Θ γP ,u+δ, then there PROOF.- From the continuity of the map x :→ IP θP (i/x) and 2 in Proposition 6.2, we have that IP θP (i/Y n )I Aγn (Y n ) →a.s. IP θP (i/Y 0 )I AγP (Y 0 ). (24) Now, taking into account that IP θP (i/·) ∈ [0, 1], we obtain that [ ] [ ] IP n IP θP (i/·)I Aγn = ν IP θP (i/Y n )I Aγn (Y n ) [ ] → ν IP θP (i/Y 0 )I AγP (Y 0 ) ] = IP [IP θP (i/·)I AγP ≥ (u + δ)IP [A γP ] , and the proof ends by applying 3 in Proposition 6.2. • Corollary 6.4 follows from Lemma 6.3 and 5 in Proposition 6.2, taking into account that in Proposition 4.4 we can take the vectors θ ∗ n as close as desired to the optimum parameters. Corollary 6.4 If IP is absolutely continuous and there exists δ > 0 such that θ P ∈ Θ γP ,u+δ, then, from an index onward, the sequence {ˆθ n } n belongs to a compact set contained in Θ. 26
Now, we are ready to prove the consistency of the procedure. PROOF OF THEOREM 4.5.- By Lemma 6.3 and the definition of ˆθ n , if n ≥ N 0 , we have IP n [ Lˆθn/γ n ] ≥ IP n [ LθP /γ n ] → IP [ L θP /γ P ] , (25) where the last convergence was stated in 5 in Proposition 6.2. Let us assume that (11) does not hold. By Corollary 6.4 the sequence {ˆθ n } n contains a subsequence such that lim n ˆθjn = θ ∗ = (π ∗ 1, ..., π ∗ k, φ ∗ 1, ..., φ ∗ k) ≠ θ P , and θ ∗ ∈ Θ. From 2 and 3 in Proposition 6.2, we have that Lˆθjn /γ jn (Y jn ) →a.s. L θ∗ /γ P (Y 0 ). (26) However, Lˆθjn /γ jn (Y jn ) ≤ I Aγjn (Y jn ) log fˆθjn (Y jn ), which is a bounded function, and we can apply Fubini’s Theorem to conclude that [ ] [ ] lim sup IP jn Lˆθjn /γ = lim sup ν Lˆθjn n jn /γ (Y jn ) ≤ ν L θ ∗ /γ n jn P (Y 0 ) = IP L θ ∗ /γ P . (27) Then, if we could prove that θ ∗ ∈ Θ γP ,u, we would have a contradiction between (27), (25) and the uniqueness of θ P . To prove this, first notice that IP ˆθjn (i/Y jn )I Aγjn (Y jn ) →a.s. IP θ ∗(i/Y 0 )I AγP (Y 0 ). (28) As the functions IP θjn (i/·) are bounded by 1, we can apply 3 in Proposition 6.2 to have ] uIP [A γP [ ] = lim uIP jn A γjn ≤ lim sup n n [ ] = lim sup ν IP ˆθjn (i/Y jn )I Aγjn (Y jn ) [ ] ≤ ν IP θ ∗(i/Y 0 )I AγP (Y 0 ) = IP IP jn [ IP θjn (i/·)I Aγjn ] [IP θ ∗(i/·)I AγP ] , so the proof is complete because Corollary 6.4 implies that π ∗ i > 0 for every i = 1, ..., k. • PROOF OF THEOREM 4.8: After our consistency results, for the analysis of the asymptotic distribution, we can assume that the γ-parameters belong to a compact subset K of ˜Γ, as well as that the θ-parameters verify the restrictions given by Θ n γ,u and belong to the set {θ : ‖θ − θ 0 ‖ < δ} for some small enough δ > 0 and large n. Now the proof parallels that given in [4] based on extended versions of the results in Section 3.2.4 in [21] to this semiparametric framework. In our case, for m θ,γ defined as in (12) the components of ṁ θ,γ := h θ,γ are those given in (14) with A γ as A. The result is then the consequence of Lemma 6.5 similar to Lemma 3.12 in [4]. From here, taking into account Proposition 4.2 and some easy computations, obtaining the asymptotic distribution given in the theorem as well as its different expressions is straightforward. • 27
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Page 29 and 30: Theorem 2.10.20 in [21] leads to th

Estimators based in adaptively trimming cells in the mixture model

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