Estimators based in adaptively trimming cells in the mixture model

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.
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