ch12sol - Chapter 12 Regression Models 12.1 The point y is...

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Chapter 12 Regression Models 12.1 The point (ˆ x , ˆ y ) is the closest if it lies on the vertex of the right triangle with vertices ( x , y ) and ( x , a + bx ). By the Pythagorean theorem, we must have x - x ) 2 + ( ˆ y - ( a + bx ) ) 2 + x - x ) 2 +(ˆ y - y ) 2 = ( x - x ) 2 + ( y - ( a + bx )) 2 . Substituting the values of ˆ x and ˆ y from (12.2.7) we obtain for the LHS above b ( y - bx - a ) 1+ b 2 2 + b 2 ( y - bx - a ) 1+ b 2 2 + b ( y - bx - a ) 1+ b 2 2 + y - bx - a ) 1+ b 2 2 = ( y - ( a + bx )) 2 b 2 + b 4 + b 2 +1 (1+ b 2 ) 2 = ( y - ( a + bx )) 2 . 12.3 a. Differentiation yields ∂f/∂ξ i = - 2( x i - ξ i ) - 2 λβ [ y i - ( α + βξ i )] set = 0 ξ i (1 + λβ 2 ) = x i - λβ ( y i - α ), which is the required solution. Also, 2 f/∂ξ 2 = 2(1 + λβ 2 ) > 0, so this is a minimum. b. Parts i), ii), and iii) are immediate. For iv) just note that D is Euclidean distance between ( x 1 , λy 1 ) and ( x 2 , λy 2 ), hence satisfies the triangle inequality. 12.5 Differentiate log L , for L in (12 . 2 . 17), to get ∂σ 2 δ log L = - n σ 2 δ + 1 2( σ 2 δ ) 2 λ 1+ ˆ β 2 n i =1 y i - α + ˆ βx i ) 2 . Set this equal to zero and solve for σ 2 δ . The answer is (12 . 2 . 18). 12.7 a. Suppressing the subscript i and the minus sign, the exponent is ( x - ξ ) 2 σ 2 δ + [ y - ( α + βξ )] 2 σ 2 = σ 2 + β 2 σ 2 δ σ 2 σ 2 δ ( ξ - k ) 2 + [ y - ( α + βx )] 2 σ 2 + β 2 σ 2 δ , where k = σ 2 x + σ 2 δ β ( y - α ) σ 2 + β 2 σ 2 δ . Thus, integrating with respect to ξ eliminates the first term. b. The resulting function must be the joint pdf of X and Y . The double integral is infinite, however. 12.9 a. From the last two equations in (12.2.19), ˆ σ 2 δ = 1 n S xx - ˆ σ 2 ξ = 1 n S xx - 1 n S xy ˆ β , which is positive only if S xx > S xy / ˆ β . Similarly, ˆ σ 2 = 1 n S yy - ˆ β 2 ˆ σ 2 ξ = 1 n S yy - ˆ β 2 1 n S xy ˆ β , which is positive only if S yy > ˆ βS xy .
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12-2 Solutions Manual for Statistical Inference b. We have from part a), ˆ σ 2 δ > 0 S xx > S xy / ˆ β and ˆ σ 2 > 0 S yy > ˆ βS xy . Furthermore, ˆ σ 2 ξ > 0 implies that S xy and ˆ β have the same sign. Thus S xx > | S xy | / | ˆ β | and S yy > | ˆ β || S xy | .
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