4.4 Tilted normal alternative
Let h(x) be a positive valued, scalar function satisfying h(0) = 1 and
1, and consider the tilted normal density
dh(0) _
dx
f(y; β, σ, κ, λ)=
1κ λ
σqκλ)φ(u) 6u3- 3u■•-u4- 6u2+3)
where u = ( Y — β)∣σ and
which is assumed to exist. Taking h(x) = |x +1| yields a density compa-
rable to the first two terms of an Edgeworth expansion. Let Fκ,λ be the
distribution corresponding to f(y;0, 1, κ, λ). Then, as κ, λ → 0, the first
four moments of Fκ,λ are (see Appendix C.3)
q(κ,λ) = EΦ
hh K(u3 — 3u)
6u2 +
3) ,
EFκ,λ(Y)=0+o(κ,λ),
EFκ,λ (Y2) = 1 + o(κ,λ), (15)
EFκ,λ (Y3) = κ + o(κ,λ),
E-. (Y4) =3 + λ + o(κ,λ),
from which κ and λ have an interpretation as skewness and (excess-)kurtosis
parameters. Our interest in this distribution lies in the fact that the score
test for κ = λ = 0 is in fact the Jarque-Bera test. Thus, under a sequence
of local alternatives
Hn : Y ~ Fn = Fκ,λ,
with κ = k/yfn and λ = l∣y∕n, the IM test with ML estimator is optimal.
We show in Appendix C.3 that, under Hn,
T →d χq2 (δ),
with non-centrality parameter
k2 l2
δ = τ + 2i
for all M-estimators of scale.
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