Testing the Information Matrix Equality with Robust Estimators

where the first line uses (1 + x)/(1 + y) ≈ 1 + x — y for x and y small,
with x = E_Fn (Y²) — 1 and y = σ²_n — 1, and the second line uses the second
moment of Student’s t, p_n/(p_n — 2). We also have that

^Fn 4   ^{- 3} = EF_n ^{(Y4)(1 - 2(σ}2 ^{- 1})) ^{- 3} + ^o(p_n 1)
^σn

3 P ²n        Σ    Σ1∖    ^, _l n n 1_λ

—  7------7√77------77 I 1--I — 3 + O(p_n )

^(pn ^{— 2)(p}n ^{— 4)} ∖    ^pn J

18 — 3∑1     , _1λ

= — ---¹ + O ( P-¹ ),                        (19)

^pn

where the first line uses (1 + x)ⁿ² ≈ 1 — 2x and the second line uses the
fourth moment of Student’s t, 3pП(p_n — 2)^{n 1}(p_n — 4)^{n 1}. Using (18) and (19),

EFn ( m 3) = ( EFσP—3) — 6 ( EFσY¹ — ɪ)
^σn              ^{σ σ}n

6 , z-H

= — ^{+ o (} P^- )

and

EF^²)! — Λ — dEFn (rh3)
∖ ^σn       /

= 2~^ ^{(4 —} ∑1 ^{— 12d)+ o(}pⁿ 1) ∙
^{2 p}n

To compute Σ₁, note that, by partial integration and using φ'(z) = —zφ(z),

E [ZPc(Z)] = j∞ zφ(z)dpc(z)

= —У Pc ( Z )( φ ( Z ) — Z ² φ ( Z )) dz

= E [Z²Pc(Z)] — bc∙

Along the same lines we get

E [Z²PC(Z)] = E [Z³p_c(Z)] — 2E [Zp'_c(Z)] ,

and

E [Z³Pc(Z)] = — 3E [Z²Pc(Z)] + E [Z⁴Pc(Z)] ,
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