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02EstimationEy - Estimation of the Response Mean Copyright...

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Estimation of the Response Mean Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 1 / 27
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The Gauss-Markov Linear Model y = X β + y is an n × 1 random vector of responses. X is an n × p matrix of constants with columns corresponding to explanatory variables. X is sometimes referred to as the design matrix . β is an unknown parameter vector in IR p . is an n × 1 random vector of errors. E ( ) = 0 and Var ( ) = σ 2 I , where σ 2 is an unknown parameter in IR + . Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 2 / 27
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The Column Space of the Design Matrix X β is a linear combination of the columns of X : X β = [ x 1 , . . . , x p ] β 1 . . . β p = β 1 x 1 + · · · + β p x p . The set of all possible linear combinations of the columns of X is called the column space of X and is denoted by C ( X ) = { Xa : a IR p } . The Gauss-Markov linear model says y is a random vector whose mean is in the column space of X and whose variance is σ 2 I for some positive real number σ 2 , i.e., E ( y ) ∈ C ( X ) and Var ( y ) = σ 2 I , σ 2 IR + . Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 3 / 27
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An Example Column Space X = 1 1 = ⇒ C ( X ) = { Xa : a IR p } = 1 1 a 1 : a 1 IR = a 1 1 1 : a 1 IR = a 1 a 1 : a 1 IR Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 4 / 27
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Another Example Column Space X = 1 0 1 0 0 1 0 1 = ⇒ C ( X ) = 1 0 1 0 0 1 0 1 a 1 a 2 : a IR 2 = a 1 1 1 0 0 + a 2 0 0 1 1 : a 1 , a 2 IR = a 1 a 1 0 0 + 0 0 a 2 a 2 : a 1 , a 2 IR = a 1 a 1 a 2 a 2 : a 1 , a 2 IR Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 5 / 27
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Different Matrices with the Same Column Space W = 1 0 1 0 0 1 0 1 X = 1 1 0 1 1 0 1 0 1 1 0 1 x ∈ C ( W ) = x = Wa for some a IR 2 = x = X 0 a for some a IR 2 = x = Xb for some b IR 3 = x ∈ C ( X ) Thus, C ( W ) ⊆ C ( X ) .
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