Estimators based in adaptively trimming cells in the mixture model

side the IF for the variances. To avoid excessive noise in the images we excluded the IF for the weights of
the component distributions. In all graphs the black curves represent the corresponding density functions
augmented 40 times.
Figure 7: IF’s for the means (blue, green and red) and the variances (cyan, yellow and magenta) of the
distributions making up a mixture of three normal distributions. The upper graphs correspond to the
mixture 1 3
(N(−5, 1) + N(0, 1) + N(5, 1)). The graph on the lower left (resp. lower right) presents the IF
for the one-step (resp. m-step) estimator for the mixture 1 4
(N(−3, 1.5) + N(0, 1.5) + 2N(3, 1.5)). The
black curves represent the corresponding density functions augmented 40 times.
To get a more accurate idea of the IF, we include the expression of the components (in π i , for i =
1, ..., k − 1, and µ i and Σ i , for i = 1, ..., k) of h θ,γ (x) as a function of θ = (π 1 , ..., π k−1 , µ 1 , ..., µ k , Σ 1 , ...Σ k )
(
∂
IP θ (i/x)
L θ/A (x) =
∂π i π i
∂
L θ/A (x) = Σ −1
i
∂µ i
∂
L θ/A (x) = 1 ∂Σ i 2
− IP )
[
θ(k/x)
IP θ (i/x)
I A (x) + IP θ − IP / ]
θ(k/x)
A c I A c(x)
π k π i π k
[
(x − µ i )IP θ (i/x) I A (x) + IP θ Σ
−1
i (x − µ i )IP θ (i/x)/A c] I A c(x),
(
Σ
−1
i (x − µ i )(x − µ i ) T Σ −1
i
+ 1 2 IP θ
[(
Σ
−1
i (x − µ i )(x − µ i ) T Σ −1
i
− Σ −1 )
i IP θ (i/x) I A (x)
− Σ −1 )
i IP θ (i/x)/A c] I A c(x).
(14)
20