PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
by the weak law of large numbers, since:
Hence,
−U ′ n(θ 0 ; x)
n
=⇒ −U ′ n(θ 0 ; x)
ni(θ 0 )
−U ′ n(θ 0 ; x)
√
ni(θ0 ) (ˆθ n − θ 0 )
=
1
n
n∑
i=1
− ∂2
∂θ 2 log f(x i; θ| θ=θ0 )
→ 1 in probability.
≈
√
ni(θ0 )(ˆθ n − θ 0 )
for large n
=⇒ √ ni(θ 0 )(ˆθ n − θ 0 )
−→
D
N(0, 1).
Remark
The preceding theory can be generalized to include vector-valued coefficients. We will
not discuss the details.
2.4 Hypothesis Tests and Confidence Intervals
Motivating Example:
Suppose X 1 , X 2 , . . . , X n are i.i.d. N(µ, σ 2 ), and consider H 0 : µ = µ 0 vs. H a : µ ≠ µ 0 .
If σ 2 is known, then the test of H 0 with significance level α is defined by the test
statistic
Z = ¯X − µ
σ/ √ n ,
and the rule, reject H 0 if |z| ≥ z(α/2).
A 100(1 − α)% CI for µ is given by
(
¯X − z(α/2) σ √ n
, ¯X + z(α/2) σ √ n
)
.
It is easy to check that the confidence interval contains all values of µ 0 that are acceptable
null hypotheses.
In general, consider a statistical problem with parameter
θ ∈ Θ 0 ∪ Θ A , where Θ 0 ∩ Θ A = φ.
We consider the null hypothesis,
H 0 : θ ∈ Θ 0
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