PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
(1) The Wald Test:
Test Statistic: W =
√
ni(ˆθ)(ˆθ − θ 0 )
Critical Region:
reject for |W | ≥ z(α/2)
(2) The Score Test:
Test Statistic: V = U(θ 0; x)
√
ni(θ0 )
Critical Region:
reject for |V | ≥ z(α/2)
(3) Likelihood-Ratio Test:
Test Statistic: G 2 = 2(l(ˆθ) − l(θ 0 ))
Example:
Critical Region: reject for G 2 ≥ χ 2 1,α
Suppose X 1 , X 2 , . . . , X n i.i.d. Po(λ), and consider H 0 : λ = λ 0 .
l(λ; x) = log
n∏ e −λ λ x i
i=1
x i !
= n(¯x log λ − λ) − log
U(λ; x) = ∂l
∂λ = n ( ¯x
λ − 1 )
=
I(λ) = ni(λ) = E
n(¯x − λ)
λ
n∏
x i !
i=1
=⇒ ˆλ = ¯x
( )
− ∂2 l
( n¯x
)
= E = n ∂λ 2 λ 2 λ
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