Testing the information matrix equality with robust estimators

The information matrix (IM) equality can be used to test for misspecification of a parametric model. We study the behavior of the IM test when the maximum-likelihood (ML) estimators used in the construction of this test are replaced with robust estimators. The latter do not suffer from the masking effect in the presence of outliers and can improve the power of the IM test. At the normal location-scale model, the IM test using the ML estimators is known as the Jarque–Bera test, and uses skewness and kurtosis to detect deviations from normality. When robust estimators are employed to test the IM equality, a robust version of the Jarque–Bera test emerges. We investigate in detail the local asymptotic power of the IM test, for various estimators and under a variety of local alternatives. For the normal regression model, it is shown by simulations under fixed alternatives that in many cases the use of robust estimators substantially increases the power of the IM test.