Further,
EFn(Y) = γnφ(0),
EFn(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 ; θn ),
and, upon replacing γn with e∣y∕n,
EFn ( m 3) = EFn |
I sq I Jλ ×---z |
EFn ( m 2) = EFn |
I I X---X sq I Jλ |
EFn ( m^ 1) = EFn |
IX ) |
_
3
_
2
_
6EFn
3EFn
(Y - βn λ
∖ σn J
2
+3=o(γn),
Y—βn = 3 Ynφ (0) + О ( Yn ),
σn
1 - dEFn ( m 3 ) = o ( Yn ),
0
b = lim ∖∕EFΓ[W [ Wm ( Y ; θn, )] = 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
31
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