# ALRLec8 - CHAPTER 5 Weights Lack of Fit and More Linear...

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1 CHAPTER 5: Weights, Lack of Fit, and More Linear Model Y = Xβ + e E(e) = 0 Var(e) = σ 2 I If the random error e’s variance is not constant, we may consider the following. Var(e) = σ 2 /w 1 0 …. σ 2 /w 2 0…. 0…. . … σ 2 /w n w 1 , …, w n are known positive numbers Var(e) = σ 2 1/w 1 ….. = σ 2 w 1 -1 1/w 2 …. w 2 ….. …. 1/w n w n = σ 2 W -1 Consider W ½ Y = W ½ X β + W ½ e where W ½ = √w 1 √w 2 …. √w n Z = Mβ + d E(d) = E(W ½ e) = W ½ E(e) = 0 Var(d) = Var( W ½ e) = W ½ Var(e) = W ½ = σ 2 I Least Square

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2 RSS(β) = (Z – Mβ) T (Z – Mβ) = (W ½ Y – W ½ Xβ) T (W ½ Y – W ½ Xβ) = (Y – Xβ) W (Y- Xβ) = Σw i (y i – x i T β) with i=1,…n β = (M T M) -1 M T Z = (X T WX) -1 X T WY Testing for Lack of Fit When y i = x i T β + e i is a correct model for data then E(σ 2 ) = σ 2 When y i = x i T β + e i is not appropriate for data, then E(σ 2 ) > σ 2 If σ 2 is known, a test for lack of fit can be obtained by comparing σ 2 with σ 2 H 0 : E(YlX) = X T β X 2 = RSS/ σ 2 = (n-p-1) σ

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## This note was uploaded on 12/12/2010 for the course STAT 425 taught by Professor Ma,p during the Fall '08 term at University of Illinois, Urbana Champaign.

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ALRLec8 - CHAPTER 5 Weights Lack of Fit and More Linear...

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