Table 1: Constants needed to compute the IM statistic T , as formulated in
(11), at the normal model for several robust estimators of scale
estimator of scale |
breakdown |
c |
bc |
d |
A |
^10% |
5.182 |
0.4476 |
0.03800 |
2.417 × 10 -4 | |
TB |
25% |
2.937 |
0.3594 |
0.1059 |
0.03163 |
50% |
1.548 |
0.1996 |
0.1925 |
0.8224 | |
MAD |
50% |
0.6745 |
0.5 |
0.2121 |
2.362 |
„ Γ - ʌ X ∏ ,ʌ ,, ʌ ʌ 1 Z- - ʌ „ X
where mn = EFn [ m ( Y ; θn )] and θn such that θ - θn = 1 ∑i=1 IF( Yi ; θ; Fn ) +
op(n-1/2). Assuming the existence of
b = lim Пптп,
n→∞
it follows that under Hn the IM test statistic T = nM'V+M is asymptoti-
cally non-central χ2 ,
T →d χq2 (δ),
with non-centrality parameter
δ = b'V+b.
In the following subsections, we derive explicit expressions for δ when
Fθ is the normal location-scale model, and Fn is a specific sequence of local
alternatives. As local alternatives we consider a contaminated normal, Stu-
dent’s t, a skewed normal, and a tilted normal. By the results of Section 3.1,
the non-centrality parameter takes the form
δ = A+1 + 1b 2 + ⅛,
6 24
where
~
b I
~
b 2
~
b 3
~
= b = Wb,
10