Testing the Information Matrix Equality with Robust Estimators



It follows that the score test statistic for testing κ = λ =0 is

S=n


μ2 + (μ4 - 3)2 ^

6 +   24


which is the Jarque-Bera statistic (see (12)).

Let κn = k/y/n and λn = l∕y∕n and Fn = Fκn ,λn. As before, βn and σn
are the solutions of (21) and (22) where expectations are now with respect
to
Fκ,λ. From (21) we have, using (24),

Eφ ψ((Z) + ψ(Z)(-βn - Z(σn - 1))}{1 + κ6n(Z3 - 3Z)

+λn ( Z4 - 6 Z 2 + 3


)   + o (βn n - 1n n ) .


(28)


Since EΦ [ψ(Z)] = 0 andψ is odd, it follows that

βn = Σ3κn + o (βn, σn - 1nn) ,                (29)

where

ς =1 E [(Z3 - 3Z)ψ(Z)]

3   6 E [ψ(Z)]

Similarly, we have, from (22),

bc


EΦ Ppc(Z) + Pc(Z)(n - Z(σn - 1))}{1 + κn(Z3


- 3Z)


)   + o (βn n - 1n n ) .

(30)


Now EΦ [ρc(Z)] = bc and ρc is even, so

σn =1+Σ4λn +(βn n - 1n n) ,             (31)

where

1 E [(Z4 - 6Z2 + 3)Pc(Z)]

24 E [ZPc(Z)]

Since

Y - βn
σn


Y (1 - (σn - 1)) - βn + o (κn, λn)

Y (1 - Σ4λn) - Σ3κn + o(κnn) ,

36



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