to admit the expansion
1n
M = n∑ m(Xi,Yi;
i=1
θ ) + ∂θmm ( Xi,Yi ; θ ))
+ op(n-1/2).
(2)
The estimator θ is B-robust (Hampel et al., 1986) when IF( ∙, ∙ ; θ; K ; Fθ ) 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 → χq2,