Testing the Information Matrix Equality with Robust Estimators



model.

In Section 2 we present the IM test with robust estimators and the
general form of the IM statistic. The IM test in the normal model, without
and with covariates, is considered in Sections 3 and 5, respectively. In
Section 4 we study the local asymptotic power of the IM test in the normal
model without covariates. Monte Carlo results are presented in Section 6.
Section 7 concludes. Technical derivations are gathered in the Appendix.

2 The IM test with robust estimators

Let (X1 ,Y1),... ,(Xn,Yn)ben independent copies of the random variables
(
X, Y ), where X has distribution K (which is left unspecified) and the distri-
bution of
Y , given X, is specified by the parametric model {Fθ | θ Θ}. Let
fθ be the conditional density corresponding to Fθ, and let sθ = — d log fθ.
The conditional IM equality can be stated as

EFθ [m(X, Y ; θ)] = 0 for almost all X,

where EFθ is the conditional expectation with respect to Fθ and m is the
vectorised lower triangular part of

sθ sθ - ∂θ sθ.

Integrating with respect to K yields the IM equality

E[m(X, Y; θ)] = 0,

where E[ ] = EKEFθ [ ]. Now let θ be an estimator of θ, sufficiently regular
to have

1n

θ — θ = n ∑IF( Xi ,Yi ; θ; k,Fθ ) + op ( n-1 / 2),           (1)

i=1

where IF is the influence function of the estimator θ, and, for

1n

M = n∑m ( Xi,Yi ; θ),
i=1



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