PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
Figure 23: Neyman-Pearson Lemma
Then f(x; µ 0)
f(x; µ a )
n∏
{
1
√ exp − 1 }
i=1 2πσ 2σ (x 2 i − µ 0 ) 2
= n∏
{
1
√ exp − 1 }
i=1 2πσ 2σ (x 2 i − µ a ) 2
{ ( n∑
)}
exp − 1 x 2
2σ 2 i − 2n¯xµ 0 + nµ 2 0
i=1
= { ( n∑
)}
exp − 1 x 2
2σ 2 i − 2n¯xµ a + nµ 2 a
i=1
{ }
1
= exp
2σ (2n¯x(µ 2 0 − µ a ) − nµ 2 0 + nµ 2 a) .
For a constant k,
f(x; µ 0 )
f(x; µ a )
⇔ (µ 0 − µ a )¯x
≤ k
≤ k ∗
⇔ ¯x ≥ c,
for a suitably chosen c (rejected when ¯x is too big).
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