Asymptotic Methods in Statistical Inference - Statistics Centre

210
• Scores test: Let ˙ 1 (θ 1 θ 2 )bethefirst elements
of ˙ (θ). Since
1
√
˙ 1 (θ 1 θ 2 ) → (0 I 11 (θ 1 θ 2 ))
we have that under ,
2 (θ 2 ) = 1 ˙ 0 1
→ 2
³
θ10 θ 2´ I−1 11
³
θ10 θ 2´
˙ 1
³
θ10 θ 2´
Finding appropriate estimates θ e 2 of θ 2 for which
2 ³
θ e 2´ →
2 can be problematic. (Exercise on
Assignment 3.)
• Note: For a 1-1 reparameterization ψ = ψ (θ),
we have I (ψ) = h J −1 i 0
I (θ) J
−1
This can be
ψ
ψ .
seen by equating the asymptotic covariance I −1 (ψ)
of √ ³ˆψ − ψ 0´
to that obtained from writing
√
³ˆψ − ψ 0´
= Jψ · √ ³ˆθ − θ 0´
+ (1), etc.