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Dr. Hackney STA Solutions pg 168

# Dr. Hackney STA Solutions pg 168 - Second Edition 10-11 b...

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Second Edition 10-11 b. The MLE of σ 2 is ˆ σ 2 μ = i ( x i - μ ) 2 /n . The information number is - d 2 d ( σ 2 ) 2 ± - n 2 log σ 2 - 1 2 ˆ σ 2 μ σ 2 ² ² ² ² σ 2 σ 2 μ = n σ 2 μ . Using the Delta method, the variance of ˆ σ μ = q ˆ σ 2 μ is ˆ σ 2 μ / 8 n , and a Wald statistic is ˆ σ μ - σ 0 q σ 2 μ / 8 n . 10 . 37 a. The log likelihood is log L = - n 2 log σ 2 - 1 2 X i ( x i - μ ) 2 2 with d = 1 σ 2 X i ( x i - μ ) = n σ 2 x - μ ) d 2 2 = - n σ 2 , so the test statistic for the score test is n σ 2 x - μ ) p σ 2 /n = n ¯ x - μ σ b. We test the equivalent hypothesis H 0 : σ 2 = σ 2 0 . The likelihood is the same as Exercise 10 . 35( b ), with ﬁrst derivative - d 2 = n σ 2 μ - σ 2 ) 2 σ 4 and expected information number E ± n (2ˆ σ 2 μ - σ 2 ) 2 σ 6 ! = n (2 σ 2 - σ 2 ) 2 σ 6 = n 2 σ 4 . The score test statistic is
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