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 ∈ Ωo is the true value. Consider now a sequence of alternatives
Hn : Y ~ Fn,
where Fn = Fωn and ωn = ω0 + e/√n. Then, under Hn,
S →d χq2 (δ),
where δ = b'V-1 b, with
b = lim Vn-Einn s (Y ; ωo ,n)
n→∞
and ωo,n solves
max EFn 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 tη (u),
where u = (y — β)/σ and tη(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) = —
∖
32
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