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
f_θ be the conditional density corresponding to F_θ, and let s_θ = — d log f_θ.
The conditional IM equality can be stated as

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

where EF_θ is the conditional expectation with respect to F_θ 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[■ ] = EKEF_θ [■ ]. 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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