02_04_2016 - Stat 151A Lecture Note Yiyang Shen RSS...

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Stat 151A Lecture Note Yiyang Shen 02/04/2016 RSS Revisited : RSS = n X i =1 ˆ e i 2 = ˆ e > ˆ e ˆ e = Y - ˆ Y = ( I - H ) Y RSS = (( I - H ) Y ) > ( I - H ) Y = Y > ( I - H ) Y X ( I - H ) = > ( I - H ) E ( RSS ) = E ( X i,j ( I - H ) ij i j ) = X ( I - H ) ij E ( i j ) = σ 2 X i ( I - H ) ii E ( i j ) = 0 i 6 = j and E ( 2 i = σ 2 ) = σ 2 ( n - p - 1) E RSS n - p - 1 = ˆ σ 2 estimated Recall Cov ( ˆ β ) = Cov (( X > X ) - 1 X > Y ) = ( X > X ) - 1 X > Cov ( Y ) X ( X > X ) - 1 = σ 2 ( X > X ) - 1 Cov ( Y ) = σ 2 I n Cov ( ˆ β ) = E RSS n - p - 1 ( X > X ) - 1 V ar ( ˆ β i ) = ˆ σ 2 ( X > X ) - 1 ii 1
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Remaining Properties of OLS Cov ( ˆ Y , ˆ e ) = Cov ( HY, ( I - H ) Y ) = HCov ( Y )( I - H ) > = HCov ( Y )( I - H ) = 2 I n ( I - H ) = σ 2 H ( I - H ) = σ 2 ( H - H 2 ) = σ 2 ( H - H ) = 0 Gauss-Markov Theorem When we estimate λ > β by λ > ˆ β LS , then λ > ˆ β LS is the B est L inear U nbiased E stimator (BLUE). If a > Y is some other linear estimator of λ > β , then V ar ( a > Y ) V ar ( λ > ˆ β LS ) where ˆ β = ( X > X ) - 1 X > Y We need V ar ( a > Y ) V ar ( λ > ˆ β LS ). E ( a > Y ) = λ > βa > = λ > β β a > X = λ > corollary (1) so: V ar ( λ > ˆ β ) = λ > [ σ 2 ( X > X ) - 1 ] λ = σ 2 a > X ( X > X ) - 1 X > a Note: V ar ( a > Y ) = σ 2 a > a V ar ( a > Y ) - V ar ( λ > ˆ β ) = σ 2 a > a - σ 2 a > X ( X > X ) - 1 X > a = σ 2 a > [ I - X ( X > X ) - 1
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