Testing the Information Matrix Equality with Robust Estimators

with A and W defined in (10) and (8). Thus, for each alternative considered,
we need to find b.

The IM test is defined by the parametric model and the estimator em-
ployed (also by the estimator of V , but this is of no concern here). It is
not oriented towards a specific alternative, hence it is called an ‘omnibus’
test for misspecification. In contrast, the score test is defined by the para-
metric model and the alternative, and is usually implemented using the ML
estimator. The fact that it is specifically designed as a test against a given
alternative, and the way the test is constructed ensure that the score test is
most p owerful against any given local alternative (Godfrey, 1990). Because
of this optimality property, it is natural to use the score test as a benchmark
for evaluating the power of the IM test. Thus, we also carry out a local power
analysis for the score test, except in the case of contamination because it
is unlikely that, in practice, the precise form of potential contamination is
known here.

In the remainder of this section, we take F_θ (y) = 1Φ y-e^, unless
otherwise stated.

4.1 Contaminated normal alternative

Consider the sequence of local alternatives

ee

Hn : Y ~ Fn = (1 - -_r ) Fθ + -^G (0 <e< 1),
nn

where G is any distribution and e is the level of contamination. We have

e lim ^M ((1 — e)F_θ + eG^
el о eV               )

with M the functional representation of M. So, b is the Gateaux derivative
of M at Fθ in the direction of G. Hence (Hampel et al., 1986)

I IF(y ; M ; Fθ ) dG (y ).

By (5), IF(y; M; F_θ) = ξ(y; θ). Therefore, under Hn,

T →^d χq² (δG)
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