Testing the Information Matrix Equality with Robust Estimators

Table 1: Constants needed to compute the IM statistic T , as formulated in
(11), at the normal model for several robust estimators of scale

„ 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 Ппт_п,
n→∞

it follows that under H_n 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
F_θ 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

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