Testing the Information Matrix Equality with Robust Estimators

where expectations are with respect to F^. From (21),

• ∕>(⅛) d -Γ∙(

^z^{(1 +}^γn ) ^-^βn

σn

dΦ(z). (23)

Now, expanding the integrands around β_n = 0, σ_n = 1, and γ_n = 0 gives

ψ ^Zz—^βn∖ = ψ(Z) + ψ/(Z) (-βn - Z(σn - 1)) + o(βn, σn - 1) (24)

∖ σ_n J

and

√z(l+γn)

∖ σn

^-^βn

) = ψ(z) + ψ(z) (ZYn - βn - z(σn - 1))

+^o(^βn^{, σ}n

(25)

and so (23)

becomes

= Eφ [ψ(Z)] - βnEφ [β'(Z)] - (σn

1)Eφ [Zψ(Z)]

¹, ^γn).

+Yn zψ ( Z ) d Φ( Z )+ O ( βn,σn

Since Eφ [ψ(Z)] = Eφ [Zψ'(Z)] = 0, it follows that β_n = γ_nΣ₂ + o(γ_n), with

_{ς =} ∕o∞ ^zψ⁽^z)^d^φ(^z)

² E [Ψ(Z)]

For σ_n it holds that

b Lpc YYd' ΓY

^z^{(1 +}^γn ) ^-^βn

σn

dΦ(z). (26)

Since (24) and (25) also hold with p_c replacing ψ, (26) becomes

^bc = ^eΦ ^[^pc⁽^z^)]^-^βn^eΦ [^p_c⁽^z)] ^-⁽^σn

■ 1)Eφ [^zp'_c(Z)]

¹, ^γn).

+Yn / Zpc(Z)dΦ(Z) + O(βn,σn-
Jo

Now Eφ [p_c(Z)] = b_c, Eφ [p'_c(Z)] = 0 and J∞ zp'_c(z)dΦ(z) = ɪE_φ [Zp_c(Z)],
so we get

^σn = ¹ + ɪ- ⁺^o⁽^γn ).