ch11sol - Chapter 11 Analysis of Variance and Regression 11.1 a The rst order Taylors series approximation is Var[g(Y[g]2 VarY =[g]2 v b If we choose

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 ) dθ = d dθ 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 ) dλ = 1 2 √ λ , Var g * ( Y ) ≈ dg * ( λ ) dλ 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 ) dθ 2 =   1 1 - θ n · 1 2 θ n · 1 n   2 = 1 4 nθ (1 - θ/n ) . Therefore Var[ g * ( Y )] ≈ dg * ( θ ) dθ 2 v ( θ ) = 1 4 n , independent of θ , that is, independent of p . c. v ( θ ) = Kθ 2 , dg * ( θ ) dθ = 1 θ and Var[ g * ( Y )] ≈ ( 1 θ ) 2 · Kθ 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 .