Testing the Information Matrix Equality with Robust Estimators

wherefrom, using (4),

Note that ξ ( Y ; θ ) does not depend on IF( Y ; /; F_θ ). Take ^ to be equivariant,
i.e. σ ( aY1 + b, ■ ■ ■ , aY_n+b ) = ∣α∣σ ( Y₁, ∙ ∙ ∙ , Y_n ), so IF( Y ; ^; F_θ ) = σ IF( u ; σ; Φ)
and

1 /    u² - 1 - 2IF(u; ^; Φ)

ξ ( Y ; θ ) = — I           u³ — 3 u

^σ \—⁴ — 5u² + 2 — 2IF(u; ^; Φ)

A straightforward calculation shows that V = σ^-4B , where B is a 3 × 3
matrix with elements Bij given by

B u = — 2 + 4ASV( ^),

B22 =6,

B33 = 46 + 4ASV( ^) — 4 E φ ( u ⁴IF),

B13 = 10 + 4ASV( ^) — 2 E φ ( u ⁴IF) = B 31,

B12 = B21 = B23 = B32 =0,

with IF = IF(u; ^; Φ) and ASV(σ) = E_φ(IF²), the asymptotic variance of
σ when σ = 1. Note that V does not depend on the estimator β that is

chosen. For a given estimator θ = (β, σ)¹, let

. ) ʌ

and write M as σ ² N with

/ ^μ2 ^{— 1}

I          .ʌ          n .ʌ

I ^μ3 ^{— 3 μ}1

ξ μ4 — 5μ2 + 2

Taking V⁺ equal to V⁺ with σ replaced by ^ yields the test statistic

T = nM '¹V⁺MM = nN^ B⁺N^.

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