Testing the Information Matrix Equality with Robust Estimators

Then, the score test statistic of H0 is

S = ns ' J^-1 s,

where

1ⁿ

s = n∑^s ( Yi ;^ω)
i=1

and J is, under H0, a consistent estimator of

J = E [ s ( Y ; ω o) s ( Y ; ω 0) ' ] ,

where ω₀ ∈ Ω_o is the true value. Consider now a sequence of alternatives

Hn : Y ~ Fn,

where F_n = F_ωn and ω_n = ω₀ + e/√n. Then, under H_n,

S →^d χq² (δ),

where δ = b'V^-1 b, with

b = lim Vn-Ein_n s (Y ; ωo ,n)
n→∞

and ωo,n solves

max EF_n log f (Y; ω).

ω∈ Ωo

C.1 Student’s t alternative

The log-density is

log f (y ; β,σ,η) = - log σ - log t_η (u),

where u = (y — β)/σ and t_η(u) is the density of a Student’s t variable with
1 /η degrees of freedom. The score function, evaluated at η = 0, is

/

s(y ; β,σ, 0) = —

∖

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