STAT100B_HW5S

STAT100B_HW5S - STAT 100B: Homework 5 Solutions Denise Tsai...

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Unformatted text preview: STAT 100B: Homework 5 Solutions Denise Tsai March 3, 2011 1. For the simplest regression model y i = βx i + i , i = 1 ,...,n , suppose x i are fixed, and E[ i ] = 0, Var[ i ] = σ 2 , and i are independent. Consider the estimator ˆ β = ∑ n i =1 w i y i / ∑ n i =1 w i x i , where w i may depends on x i , but not on y i . (a) Find E[ ˆ β ] and Var[ ˆ β ]. Solution: E( ˆ β ) = E ∑ n i =1 w i y i ∑ n i =1 w i x i = ∑ n i =1 E[ w i ( βx i + i )] ∑ n i =1 w i x i = ∑ n i =1 ( w i βx i + 0) ∑ n i =1 w i x i since E( w i i ) = w i E( i ) = 0 = β ( ∑ n i =1 w i x i ) ∑ n i =1 w i x i = β Var( ˆ β ) = Var ∑ n i =1 w i y i ∑ n i =1 w i x i = ∑ n i =1 Var[ w i ( βx i + i )] ( ∑ n i =1 w i x i ) 2 = ∑ n i =1 (0 + w 2 i σ 2 ) ∑ n i =1 w i x i since Var( w i i ) = w 2 i Var( i ) = w 2 i σ 2 = ∑ n i =1 w 2 i σ 2 ∑ n i =1 w i x i 1 (b) Show that Var[ ˆ β ] is minimized when w i ∝ x i . Calculate the minimum. Solution: Since σ 2 > 0, min ∑ n i =1 w 2 i σ 2 ∑ n i =1 w i x i = min ∑ n i =1 w 2 i ∑ n i =1 w i x i . Then, ∑ n i =1 w 2 i ∑ n i =1 w i x i = | ~w | 2 h ~w,~x i 2 = | ~w | 2 | ~w | 2 | ~x | 2 cos 2 θ = 1 | ~x | 2 cos 2 θ ≥ 1 | ~x | 2 Var( β ) is minimized when cos 2 θ = 1 ⇒ θ = 0 or θ = π ⇒ ~w and ~x are on the same line. ⇒ ~w ∝ ~x ( w i ∝ x i ) 2. For the simple regression y i = β + β 1 x i + i , i = 1 ,...,n , define the sample variance of ( x i ,i = 1 ,...,n ) as V x = ∑ n...
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This note was uploaded on 09/20/2011 for the course STAT 100B taught by Professor Wu during the Winter '11 term at UCLA.

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STAT100B_HW5S - STAT 100B: Homework 5 Solutions Denise Tsai...

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