Testing the Information Matrix Equality with Robust Estimators



from which an alternative expression for Σ1 follows,

Σ1 =


E[(Z4 - 2Z2) ρc(Z)] - bc
E [Z2ρc(Z)] - bc

From (9),

Eφ [(Z4 - 6Z2 + 3)(Z2 - 1 - 2IF)]

=        Eφ ((Z4 - 6Z2 + 3)2)

= 24Eφ [(Z4 - 6Z2 + 3)(Z2 - 1 - 2IF)]

=    1 EΦ [(Z4 - 6Z2) Pc(Z)] +3bc

(20)


=    12 E [Z2ρc(Z)] - bc ,

from which it is straightforward that 4 - 12d = Σ1, and thus EFn (m 1) =
o(p-1). Replacing pn with √ne, we obtain

0 0

b = lim E'n-. [ Wm ( Y ; θn )] = el 0
n→∞

6

and

δ = bV+b =3 e 2.

B.2 Skewed normal alternative

Let γn = e/y/n and Fn = FγSn. Let β be estimated by an M-estimator of
location (e.g. the median). Then
βn and σn are the solutions of
where
ψ is an odd function, non-decreasing, not identically zero, and differ-
entiable a.e., and

n )1
σn


=0,


(21)


E    (-½) 1= bc,                 (22)

σn

29



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