Testing the Information Matrix Equality with Robust Estimators

δG = e² Eg [ξ ( Y ; θ)] ' V⁺Eg [ξ ( Y ; θ)].

The expression for the non-centrality parameter is valid for any Fθ and G,
provided that EG[ξ(y; θ)] exists. Note that, for G = F_θ, δG = 0 as it should
be.

When Fθ is the normal distribution, then

δG = e² A⁺ (E_σξι)2 + 1 (EGξ'2)² + 24 (Egёз)2

where

ξι = u² — 1 — 2IF — dξ3,
ξ₂ = u³ — 3 u,
ξ₃ = u⁴ — 6 u² + 3,

IF = IF(u; σ; Φ) and u = Y^-β. We see from (14) that, for any G, δG is
minimal when σ is the ML estimator, because then A = d = ξι = 0, which
makes the first term in brackets vanish, and because the second and third
term are independent of σ. When σ is not the ML estimator, the first
term vanishes only if EGξ₁ = 0. Thus, replacing the ML estimator with
another estimator has two opposite effects on power: it increases (actually,
non-decreases) the non-centrality parameter, but also increases the degrees
of freedom from 2 to 3.

Taking G =∆_x , where ∆_x is the Dirac measure with all mass at x
(representing outliers in the data at x), the non-centrality parameter is

δχ = e ²( ξ ( x ; θ )) 'V⁺ξ ( x ; θ ),

while for G = 11 (∆_x + ∆_-x) (representing symmetric outliers), it is

e²

δ-x,x = Y (ξ(X; θ) + ξ(—X; θ)) V ⁺(ξ(X; θ) + ξ(—x; θ)).

Figure 1 gives the non-centrality parameter for G =∆_x and the three es-
timators discussed above (ML, MAD, and TB with 25% breakdown point).
The non-centrality parameter for G = 2(∆_x + ∆_-x) is plotted in Figure 2.

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