Solution to tutorial 10

# Solution to tutorial 10 - Tutorial 10 1. Derive the...

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Tutorial 10 1. Derive the weighted least square normal equations for Ftting a simple linear regression func- tion when σ 2 i = kX i ,where k> 0isaconstant . Let Q w ( b 0 ,b 1 )= n X i =1 1 i ( Y i - b 0 - b 1 X i ) 2 and ∂Q w ( b 0 1 ) ∂b 0 = - 2 n X i =1 1 i ( Y i - b 0 - b 1 X i ) w ( b 0 1 ) 1 = - 2 n X i =1 1 k ( Y i - b 0 - b 1 X i ) . Setting the derivatives equal to zero, simplifying, n X i =1 Y i X i - b 0 n X i =1 1 X i - nb 1 =0 n X i =1 Y i - nb 0 - b 1 n X i =1 X i . we have b 1 = n i =1 Y i X i - ¯ Y n i =1 1 X i n - ¯ X n i =1 1 X i b 0 = ¯ Y - b 1 ¯ X. where ¯ X = n X i =1 X i /n, ¯ Y = n X i =1 Y i /n, 2. ±or linear regression model Y i = β 0 + β 1 X i 1 + ... + β p X ip + ε i ,i =1 , ..., n. with Var ( ε 1 . . . ε n σ 2 1 0 ... 0 0 σ 2 2 ... 0 ··· 00 ... σ 2 n (a) If the LSE, b , is used, is the estimator unbiased? what is the variance of the estimated coeﬃcients, ( b ). (b) with w i 2 i , derive the weighted least square estimator b w , and calculate ( b w ) 1

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(a) The LSE is b =( X 0 X ) 1 X 0 Y , We have E ( b )=( X 0 X ) 1 X 0 E ( Y X 0 X ) 1 X 0 X β = β SO, it is still unbiased. Because
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## This note was uploaded on 10/04/2010 for the course STAT ST3131 taught by Professor Xiayingcun during the Fall '09 term at National University of Singapore.

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Solution to tutorial 10 - Tutorial 10 1. Derive the...

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