Testing the Information Matrix Equality with Robust Estimators

to admit the expansion

1ⁿ

M = n∑ m(Xi,Yi;

i=1

θ ) + ∂θm^m ( Xi,Yi ; θ ))

+ op(n^-¹^/²).

(2)

The estimator θ is B-robust (Hampel et al., 1986) when IF( ∙, ∙ ; θ; K ; F_θ ) is
bounded. Assuming the existence of

D ( θ ) = E [ ɪ m ( X,Y ; θ )],

we have

1n∂

-∑ _dθ m ( Xi,Yi ;^θ ) → D (^θ ) ■ (3)

i=1

Now, let

ξ ( X, Y ; θ ) = m ( X, Y ; θ ) + D ( θ )IF( X, Y ; θ; K, Fθ ). (4)

Then, combining (1)-(4),

1ⁿ

M^ = n ∑ ξ ( Xi, Yi ; θ ) + Op ( n^-¹ / 2). (5)

i=1

So we obtain

_x7^∖l → N(0 ,V ),

with

V = E [ξ (X,Y ; θ) ξ (X,Y ; θ) '].

Let V⁺ be a consistent estimator of V⁺, the Moore-Penrose inverse of V,
and define the test statistic

T = nM '¹V⁺M^.

Then, if the parametric model is correctly specified,

d2
T → χ_q²,