Testing the Information Matrix Equality with Robust Estimators

from which an alternative expression for Σ1 follows,

E[(Z⁴ - 2Z2) ρc(Z)] - bc
E [Z2ρc(Z)] - bc

From (9),

Eφ [(Z⁴ - 6Z2 + 3)(Z2 - 1 - 2IF)]

=        Eφ ((Z⁴ - 6Z2 + 3)2)

= 24Eφ [(Z⁴ - 6Z2 + 3)(Z2 - 1 - 2IF)]

₌    1 EΦ [(Z⁴ - 6Z2) Pc(Z)] +3bc

=    12 E [Z2ρc(Z)] - bc ^,

from which it is straightforward that 4 - 12d = Σ₁, and thus EFn (m ₁) =
o(p-¹). Replacing p_n with √n∣e, we obtain

0 ⁰

b = lim ∖E∣'_n-. [ Wm ( Y ; θ_n )] = el 0
n→∞

6

and

δ = bV⁺b =3 e 2.

B.2 Skewed normal alternative

Let γ_n = e/y/n and F_n = FγSn. Let β be estimated by an M-estimator of
location (e.g. the median). Then βn and σn are the solutions of
where ψ is an odd function, non-decreasing, not identically zero, and differ-
entiable a.e., and

E    (-½) 1= bc,                 (22)

σn

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