Testing the Information Matrix Equality with Robust Estimators

we obtain

EFn

Y - βn
σn

-Σ3κ_n + o(κ_n,λ_n),

_f 17 y - βn V
EFn [к σ .

f 17 y - βn 73
EFn [k~τr).
P ^ff Y - βn y'
EFn l σ

1- 2Σ4λ_n + o(κ_n,λ_n),

κ_n - 3Σ3κ_n + o(κ_n, λ_n),

3+λ_n - 12Σ4λ_n + o(κ_n,λ_n).

Letting ( m 1 ,m 2 ,m 3) = Wm ( Y ; θ_n ), this results in

EFn ^(m 1) = ^-λn(^2ς4 + ^d) + ^o(κn^{, λ}n)

EFn (m2) = Kn + O(K_n, λn)

EFn (m¹3) = λn + O(Kn, λn)

Note that, from (20),

_    1 E [(Z⁴ - 6Z2 + 3)Pc(Z)]

=    12       E [ZPc(Z)]

= -2Σ4,

and thus EF_n [rn ₁] = o(κ_n, λ_n). Since κ_n = k/yfn and λ_n = l∕√n, we obtain

b = lim nEE_Fn [ Wm ( Y ; θn )] =
n→∞

from which it is straightforward that

₊ k² l²

^δ=_bv^+b=_⁺-.

Acknowledgements

Financial support from the Flemish Fund for Scientific Research (grant
G.0366.01) is gratefully acknowledged.

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