Estimators based in adaptively trimming cells in the mixture model

But, if we keep the uniqueness of θ P and we apply the argument in Theorem 4.5 to each of the
subsequences for which the trimmed k-means converge, we would have that, if ω ∈ Ω 0 , then every
subsequence of {ˆθ n } contains a further subsequence which converges to θ P , and, in consequence, the
whole sequence converges to θ P .
•
The reasoning leading to the consistency of the procedure is only based on the (a.s.) weak convergence
of the sample probability measures to IP . Therefore, the same proof works to prove its continuity with
respect to the weak convergence:
Corollary 4.7 (Qualitative Robustness) Let IP = IP θ0 for some θ 0 ∈ Θ and assume that θ 0 ∈
Θ γP ,u+δ, for some δ > 0. Let {Q n } be a sequence of probability measures that converges in distribution
to IP . Then lim n θ Qn = θ 0 .
To obtain the asymptotic law of the estimator we resort to the empirical processes theory, as developed
in Van der Waart and Wellner [21]. We will take advantage of the parametric nature of the trimmed sets
under consideration. To this end, let ˜Γ
(
:= IR d) k (
× M
+ k (
d×d)
× IR
+ ) k
indexing the sets constituted
by the union of k ellipsoids. For γ = (m 1 , ..., m k , Σ 1 , ..., Σ k , r 1 , ..., r k ) ∈ ˜Γ, let A γ := ⋃ k
i=1 {x ∈ IRd :
(x − m i ) T Σ −1
i (x − m i ) ≤ r i }.
We can use arguments of the Empirical Process Theory for the family of functions
G Λ :=
{m θ,γ := I Aγ log (f θ ) + I A c
γ
log ( ( )) }
IP θ A
c
γ , (θ, γ) ∈ Λ , (12)
and their derivatives with respect to θ:
( ) ( ∂ ∂
h θ,γ := I Aγ
∂θ log (f θ) + I A c
γ
∂θ log ( ( )) )
IP θ A
c
γ
where Λ is a suitable subset of Θ × ˜Γ.
As noted in [4] the extension of the argmax arguments of the Empirical Processes Theory to this
semiparametric model is an easy fact through the extensions of the results of Section 3.2.4 in [21] given
by Theorem 5.2 and Lemma 5.3 in [4]. From these extended statements the results will arise from that
work after some algebra on Donsker classes based on the theory included in [21] as we will prove in
Lemma 6.5.
Theorem 4.8 (Asymptotic distribution) Let IP = IP θ0 , for some θ 0 ∈ Θ, and γ 0 ∈ ˜Γ. If θ 0 ∈
Θ γ0,u+δ for some δ > 0 and {γ n } n
is a sequence (possibly random) in ˜Γ such that γ n → γ 0 a.s. then the
sequence
{ˆθn (γ n )}
of estimators based on the sets A γ n
verifies
n
⎛ (
√ )
) ⎞ −1
∂
n
(ˆθn (γ n ) − θ 0 → w N ⎝0,
∂θ ∣ IP θ0 h θ,γ0
⎠ .
θ=θ0
The asymptotic covariance matrix can also be expressed as
(IP θ0
(
(h θ0,γ 0
) (h θ0,γ 0
) T )) −1
.
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