Testing the Information Matrix Equality with Robust Estimators

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

1κ             λ

σqκλ)^φ(^u) 6^u3^{- 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 F_κ,λ are (see Appendix C.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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