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Dr. Hackney STA Solutions pg 146

Dr. Hackney STA Solutions pg 146 - 9-4 Solutions Manual for...

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9-4 Solutions Manual for Statistical Inference The LRT statistic is λ ( x ) = ˆ a ˆ θ a 0 ˆ θ 0 n/ 2 e - 1 2 a 0 ˆ θ 0 Σ( x i - ˆ θ 0 ) 2 e - 1 a ˆ θ Σ( x i - ˆ θ ) 2 = 1 2 πa 0 ˆ θ 0 n/ 2 e n/ 2 e - 1 2 a 0 ˆ θ 0 Σ( x i - ˆ θ 0 ) 2 The rejection region of a size α test is { x : λ ( x ) c α } , and a 1 - α confidence set is { a 0 : λ ( x ) c α } . b. Using the results of Exercise 8.8b, the restricted MLE (for a = a 0 ) is found by solving - a 0 θ 2 + [ˆ σ 2 + (¯ x - θ ) 2 ] + θ x - θ ) = 0 , yielding the MLE ˆ θ R = ¯ x + ¯ x + 4 a 0 σ 2 + ¯ x 2 ) / 2 a 0 . The unrestricted MLEs are ˆ θ = ¯ x and ˆ a = 1 n ¯ x 2 n i =1 ( x i - ¯ x ) 2 = ˆ σ 2 ¯ x 2 , yielding the LRT statistic λ ( x ) = ˆ σ/ ˆ θ R n e ( n/ 2) - Σ( x i - ˆ θ R ) 2 / (2 ˆ θ R ) . The rejection region of a size α test is { x : λ ( x ) c α } , and a 1 - α confidence set is { a 0 : λ ( x ) c α } . 9.9 Let Z 1 , . . . , Z n be iid with pdf f ( z ). a. For X i f ( x - μ ), ( X 1 , . . . , X n ) ( Z 1 + μ, . . . , Z n + μ ), and ¯ X - μ Z + μ - μ = ¯ Z . The distribution of ¯ Z does not depend on μ .
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