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ch11sol - Chapter 11 Analysis of Variance and Regression...

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Chapter 11 Analysis of Variance and Regression 11.1 a. The first order Taylor’s series approximation is Var[ g ( Y )] [ g ( θ )] 2 · Var Y = [ g ( θ )] 2 · v ( θ ) . b. If we choose g ( y ) = g * ( y ) = y a 1 v ( x ) dx , then dg * ( θ ) = d θ a 1 v ( x ) dx = 1 v ( θ ) , by the Fundamental Theorem of Calculus. Then, for any θ , Var[ g * ( Y )] 1 v ( θ ) 2 v ( θ ) = 1 . 11.2 a. v ( λ ) = λ , g * ( y ) = y , dg * ( λ ) = 1 2 λ , Var g * ( Y ) dg * ( λ ) 2 · v ( λ ) = 1 / 4, independent of λ . b. To use the Taylor’s series approximation, we need to express everything in terms of θ = E Y = np . Then v ( θ ) = θ (1 - θ/n ) and dg * ( θ ) 2 = 1 1 - θ n · 1 2 θ n · 1 n 2 = 1 4 (1 - θ/n ) . Therefore Var[ g * ( Y )] dg * ( θ ) 2 v ( θ ) = 1 4 n , independent of θ , that is, independent of p . c. v ( θ ) = 2 , dg * ( θ ) = 1 θ and Var[ g * ( Y )] ( 1 θ ) 2 · 2 = K , independent of θ . 11.3 a. g * λ ( y ) is clearly continuous with the possible exception of λ = 0. For that value use l’Hˆ opital’s rule to get lim λ 0 y λ - 1 λ = lim λ 0 (log y ) y λ 1 = log y. b. From Exercise 11.1, we want to find v ( λ ) that satisfies y λ - 1 λ = y a 1 v ( x ) dx. Taking derivatives d dy y λ - 1 λ = y λ - 1 = d dy y a 1 v ( x ) dx = 1 v ( y ) .
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11-2 Solutions Manual for Statistical Inference Thus v ( y ) = y - 2( λ - 1) . From Exercise 11.1, Var y λ - 1 λ d dy θ λ - 1 λ 2 v ( θ ) = θ 2( λ - 1) θ - 2( λ - 1) = 1 . Note: If λ = 1 / 2, v ( θ ) = θ , which agrees with Exercise 11.2(a). If λ = 1 then v ( θ ) = θ 2 , which agrees with Exercise 11.2(c). 11.5 For the model Y ij = μ + τ i + ε ij , i = 1 , . . . , k, j = 1 , . . . , n i , take k = 2. The two parameter configurations ( μ, τ 1 , τ 2 ) = (10 , 5 , 2) ( μ, τ 1 , τ 2 ) = (7 , 8 , 5) , have the same values for μ + τ 1 and μ + τ 2 , so they give the same distributions for Y 1 and Y 2 . 11.6 a. Under the ANOVA assumptions Y ij = θ i + ij , where ij independent n(0 , σ 2 ), so Y ij independent n( θ i , σ 2 ). Therefore the sample pdf is k i =1 n i j =1 (2 πσ 2 ) - 1 / 2 e - ( y ij - θ i ) 2 2 σ 2 = (2 πσ 2 ) - Σ n i / 2 exp - 1 2 σ 2 k i =1 n i j =1 ( y ij - θ i ) 2 = (2 πσ 2 ) - Σ n i / 2 exp - 1 2 σ 2 k i =1 n i θ 2 i × exp - 1 2 σ 2 i j y 2 ij + 2 2 σ 2 k i =1 θ i n i ¯ Y i · . Therefore, by the Factorization Theorem, ¯ Y 1 · , ¯ Y 2 · , . . . , ¯ Y k · , i j Y 2 ij is jointly sufficient for ( θ 1 , . . . , θ k , σ 2 ) . Since ( ¯ Y 1 · , . . . , ¯ Y k · , S 2 p ) is a 1-to-1 function of this vector, ( ¯ Y 1 · , . . . , ¯ Y k · , S 2 p ) is also jointly sufficient. b. We can write (2 πσ 2 ) - Σ n i / 2 exp - 1 2 σ 2 k i =1 n i j =1 ( y ij - θ i ) 2 = (2 πσ 2 ) - Σ n i / 2 exp - 1 2 σ 2 k i =1 n i j =1 ([ y ij - ¯ y i · ] + [¯ y i · - θ i ]) 2 = (2 πσ 2 ) - Σ n i / 2 exp - 1 2 σ 2 k i =1 n i j =1 [ y ij - ¯ y i · ] 2 exp - 1 2 σ 2 k i =1 n i y i · - θ i ] 2 , so, by the Factorization Theorem, ¯ Y i · , i = 1 , . . . , n , is independent of Y ij - ¯ Y i · , j = 1 , . . . , n i , so S 2 p is independent of each ¯ Y i · . c. Just identify n i ¯ Y i · with X i and redefine θ i as n i θ i .
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Second Edition 11-3 11.7 Let U i = ¯ Y i · - θ i . Then k i =1 n i [( ¯ Y i · - ¯ ¯ Y ) - ( θ i - ¯ θ )] 2 = k i =1 n i ( U i - ¯ U ) 2 .
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