PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
See Figure 22 and observe α ≤ α ∗
∫
∫
=⇒ f(x; θ 0 )dx 1 . . . dx n ≤
D
C
f(x; θ 0 )dx 1 . . . dx n
∫
∫
=⇒ f(x; θ 0 )dx 1 . . . dx n ≤ f(x; θ 0 )dx 1 . . . dx n
D 2 C 2
∫
∫
=⇒ (1 − β ∗ ) − (1 − β) = f(x; θ a )dx 1 . . . dx n −
C
D
f(x; θ a )dx 1 . . . dx n
∫
∫
∴ (1 − β ∗ ) − (1 − β) = f(x; θ a )dx 1 . . . dx n + f(x; θ a )dx 1 . . . dx n
C 1 C 2
∫
∫
− f(x; θ a )dx 1 . . . dx n − f(x; θ a )dx 1 . . . dx n
C 1 D 2
=
≥
(C 1 ∪ D 2 )
∫
∫
f(x; θ a )dx 1 . . . dx n − f(x; θ a )dx 1 . . . dx n
C 2 D 2 1 ∫
f(x; θ 0 )dx 1 . . . dx n − 1 ∫
f(x; θ 0 )dx 1 . . . dx n
k C 2
k D 2
≥ 0, as required.
See Figure 23.
Moreover, equality is achieved only if
D 2 is empty (= φ).
Example
Suppose X 1 , X 2 , . . . , X n are i.i.d. N(µ, σ 2 ), σ 2 given, and consider
H 0 : µ = µ 0 vs. H a : µ = µ a , µ a > µ 0
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