Testing the Information Matrix Equality with Robust Estimators

Jarque-Bera test. We have shown that, under local contamination, the ML
estimator in fact minimizes the non-centrality parameter that appears in the
limiting χ² distribution of the test statistic. Under some other alternatives
(Student’s t, skewed normal, tilted normal), the non-centrality parameter
was found to be the same, whether using the ML estimator or robust esti-
mators. In the regression model, the IM test with robust estimators clearly
dominates the IM test with ML estimator in the presence of vertical out-
liers. Somewhat to our surprise, only minor differences between the tests
were found under a Cauchy alternative.

The use of robust estimators makes the parameter estimates much less
sensitive to outlying observations than when the ML estimator is used. As a
result, such observations are more easily recognised as outliers, and outliers
are in this context considered as evidence against the model. This intuition
is supported by the analytical results concerning local contamination, for
an arbitrary contaminating distribution, and by simulation results in the
regression case.

We have focussed on the normal location-scale model and the regression
model. The potential of using robust estimators in connection with the IM
test in other models remains to be explored.

Appendix A

Computation of T using TB or MAD estimator

To compute T using the TB estimator, let, for k even,

^c k        2^k/2    k +1 k +1 c²

^νc(k) = Jc u ^dφ(^u) = ∏^γΓ (ʃ J P Ç-_r-, - J ,

with Γ the gamma and P the incomplete gamma function. Now,

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