ch9sol - Chapter 9 Interval Estimation 9.1 Denote A = x L x...

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Unformatted text preview: Chapter 9 Interval Estimation 9.1 Denote A = { x : L ( x ) ≤ θ } and B = { x : U ( x ) ≥ θ } . Then A ∩ B = { x : L ( x ) ≤ θ ≤ U ( x ) } and 1 ≥ P { A ∪ B } = P { L ( X ) ≤ θ or θ ≤ U ( X ) } ≥ P { L ( X ) ≤ θ or θ ≤ L ( X ) } = 1, since L ( x ) ≤ U ( x ). Therefore, P ( A ∩ B ) = P ( A )+ P ( B )- P ( A ∪ B ) = 1- α 1 +1- α 2- 1 = 1- α 1- α 2 . 9.3 a. The MLE of β is X ( n ) = max X i . Since β is a scale parameter, X ( n ) /β is a pivot, and . 05 = P β ( X ( n ) /β ≤ c ) = P β (all X i ≤ cβ ) = cβ β α n = c α n implies c = . 05 1 /α n . Thus, . 95 = P β ( X ( n ) /β > c ) = P β ( X ( n ) /c > β ), and { β : β < X ( n ) / ( . 05 1 /α n ) } is a 95% upper confidence limit for β . b. From 7.10, ˆ α = 12 . 59 and X ( n ) = 25. So the confidence interval is (0 , 25 / [ . 05 1 / (12 . 59 · 14) ]) = (0 , 25 . 43). 9.4 a. λ ( x,y ) = sup λ = λ L ( σ 2 X ,σ 2 Y x,y ) sup λ ∈ (0 , + ∞ ) L ( σ 2 X ,σ 2 Y | x,y ) The unrestricted MLEs of σ 2 X and σ 2 Y are ˆ σ 2 X = Σ X 2 i n and ˆ σ 2 Y = Σ Y 2 i m , as usual. Under the restriction, λ = λ , σ 2 Y = λ σ 2 X , and L ( σ 2 X ,λ σ 2 X x,y ) = ( 2 πσ 2 X )- n/ 2 ( 2 πλ σ 2 X )- m/ 2 e- Σ x 2 i / (2 σ 2 X ) · e- Σ y 2 i / (2 λ σ 2 X ) = ( 2 πσ 2 X )- ( m + n ) / 2 λ- m/ 2 e- ( λ Σ x 2 i +Σ y 2 i ) / (2 λ σ 2 X ) Differentiating the log likelihood gives d log L d ( σ 2 X ) 2 = d dσ 2 X- m + n 2 log σ 2 X- m + n 2 log (2 π )- m 2 log λ- λ Σ x 2 i + Σ y 2 i 2 λ σ 2 X =- m + n 2 ( σ 2 X )- 1 + λ Σ x 2 i + Σ y 2 i 2 λ ( σ 2 X )- 2 set = which implies ˆ σ 2 = λ Σ x 2 i + Σ y 2 i λ ( m + n ) . To see this is a maximum, check the second derivative: d 2 log L d ( σ 2 X ) 2 = m + n 2 ( σ 2 X )- 2- 1 λ ( λ Σ x 2 i + Σ y 2 i ) ( σ 2 X )- 3 σ 2 X =ˆ σ 2 =- m + n 2 (ˆ σ 2 )- 2 < , 9-2 Solutions Manual for Statistical Inference therefore ˆ σ 2 is the MLE. The LRT statistic is ( ˆ σ 2 X ) n/ 2 ( ˆ σ 2 Y ) m/ 2 λ m/ 2 (ˆ σ 2 ) ( m + n ) / 2 , and the test is: Reject H if λ ( x,y ) < k , where k is chosen to give the test size α . b. Under H , ∑ Y 2 i / ( λ σ 2 X ) ∼ χ 2 m and ∑ X 2 i /σ 2 X ∼ χ 2 n , independent. Also, we can write λ ( X,Y ) = 1 n m + n + (Σ Y 2 i /λ σ 2 X ) /m (Σ X 2 i /σ 2 X ) /n · m m + n n/ 2 1 m m + n + (Σ X 2 i /σ 2 X ) /n (Σ Y 2 i /λ σ 2 X ) /m · n m + n m/ 2 = " 1 n n + m + m m + n F # n/ 2 " 1 m m + n + n m + n F- 1 # m/ 2 where F = Σ Y 2 i /λ m Σ X 2 i /n ∼ F m,n under H . The rejection region is ( x,y ): 1 h n n + m + m m + n F i n/ 2 · 1 h m m + n + n m + n F- 1 i m/ 2 < c α where c α is chosen to satisfy P n n + m + m m + n F- n/ 2 m n + m + n m + n F- 1- m/ 2 < c α = α....
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ch9sol - Chapter 9 Interval Estimation 9.1 Denote A = x L x...

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