Testing the Information Matrix Equality with Robust Estimators

EFn(Y) = γnφ(0),
EF_n(Y²)=1+γn + o(γn),
EFn(Y ³) = 6γnφ(0) + o(γn),
EFn(Y⁴)=3+6γn+o(γn).

Therefore, letting ( m 1 ,m 2 ,m 3) ' = Wm ( Y ; θ_n ),
and, upon replacing γ_n with e∣y∕n,

0

b = lim ∖∕EFΓ[W [ Wm ( Y ; θ_n, )] = e   3 φ (0)

n→∞                      0

Hence bV⁺b = -⅛ e2.

Appendix C

Local asymptotic power of score test

First, we review briefly how the local asymptotic power of the score test
against specified alternatives can be defined. By an appropriate extension
of f ( ■ ; ■ ), let the density under the alternative be f ( y ; ω ), depending on
an extended parameter ω, and let s ( y ; ω ) = — ∂ω log f ( У ; ω ). Write the
null hypothesis as H₀ : ω ∈ Ω₀, where Ω₀ is a restricted parameter space
(essentially, Θ). Let ω be the restricted ML estimator (essentially, θ), i.e. ω
solves

n

max    log f (Yi; ω).

ω∈Ωo

i=1

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