Dr. Hackney STA Solutions pg 136

Dr. Hackney STA Solutions pg 136 - Second Edition 8-15 8.37...

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Unformatted text preview: Second Edition 8-15 8.37 a. P ( ¯ X > θ + z α σ/ √ n | θ ) = P ( ( ¯ X- θ ) / ( σ/ √ n ) > z α | θ ) = P ( Z > z α ) = α , where Z ∼ n(0 , 1). Because ¯ x is the unrestricted MLE, and the restricted MLE is θ if ¯ x > θ , the LRT statistic is, for ¯ x ≥ θ λ ( x ) = (2 πσ 2 )- n/ 2 e- Σ i ( x i- θ ) 2 / 2 σ 2 (2 πσ 2 )- n/ 2 e- Σ i ( x i- ¯ x ) 2 / 2 σ 2 = e- [ n (¯ x- θ ) 2 +( n- 1) s 2 ] ] . 2 σ 2 e- ( n- 1) s 2 / 2 σ 2 = e- n (¯ x- θ ) 2 / 2 σ 2 . and the LRT statistic is 1 for ¯ x < θ . Thus, rejecting if λ < c is equivalent to rejecting if (¯ x- θ ) / ( σ/ √ n ) > c (as long as c < 1 – see Exercise 8.24). b. The test is UMP by the Karlin-Rubin Theorem. c. P ( ¯ X > θ + t n- 1 ,α S/ √ n | θ = θ ) = P ( T n- 1 > t n- 1 ,α ) = α , when T n- 1 is a Student’s t random variable with n- 1 degrees of freedom. If we define ˆ σ 2 = 1 n ∑ ( x i- ¯ x ) 2 and ˆ σ 2 = 1 n ∑ ( x i- θ ) 2 , then for ¯ x ≥ θ the LRT statistic is...
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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