PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
and the alternative hypothesis
H A : θ ∈ Θ A .
The hypothesis testing set up can be represented as:
Test Result
Actual Status
H 0 true H A true
Accept H 0
√
type II
error
(β)
Reject H 0
type I
error
√
(α)
We would like both the type I and type II error rates to be as small as possible.
However, these results conflict with each other. To reduce the type I error rate we
need to “make it harder to reject H 0 ”. To reduce the type II error rate we need to
“make it easier to reject H 0 ”.
The standard (Neyman-Pearson) approach to hypothesis testing is to control the type
I error rate at a “small” value α and then use a test that makes the type II error as
small as possible.
The equivalence between the confidence intervals and hypothesis tests can be formulated
as follows: Recall that a 100(1 − α)% CI for θ is a random interval, (L, U) with
the property
P ((L, U) ∋ θ) = 1 − α.
It is easy to check that the test defined by rule:
“Accept H 0 : θ = θ 0 iff θ 0 ∈ (L, U)” has significance level α.
α = P (reject H 0 |H 0 true)
β = P (retain H 0 |H A true)
1 − β = power = P (reject H 0 |H A true), which is what we want.
Conversely, given a hypothesis test H 0 : θ = θ 0 with significance level α, it can be
proved that the set {θ 0 : H 0 : θ = θ 0 is accepted} is a 100(1 − α)% confidence region
for θ.
Large Sample Tests and Confidence Intervals
Consider a statistical problem with data x 1 , . . . , x n , log-likelihood l(θ; x), score U(θ; x)
and information i(θ).
Consider also a hypothesis H 0 : θ = θ 0 . The following three tests are often considered:
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