temp - Theorem: Under the conditions of the previous...

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Theorem: Under the conditions of the previous theorem, and where var( e ) = var( y ) = σ 2 I , the variance of λ T ˆ β is unique, and is given by var( λ T ˆ β ) = σ 2 λ T ( X T X ) - λ , where ( X T X ) - is any generalized inverse of X T X . Proof: var( λ T ˆ β ) = λ T var(( X T X ) - X T y ) λ = λ T ( X T X ) - X T σ 2 IX { ( X T X ) - } T λ = σ 2 λ T ( X T X ) - X T X | {z } = λ T { ( X T X ) - } T λ = σ 2 λ T { ( X T X ) - } T λ = σ 2 a T X { ( X T X ) - } T X T a (for some a ) = σ 2 a T X ( X T X ) - X T a = σ 2 λ T ( X T X ) - λ . Uniqueness: since λ T β is estimable λ = X T a for some a . Therefore, var( λ T ˆ β ) = σ 2 λ T ( X T X ) - λ = σ 2 a T X ( X T X ) - X T a = σ 2 a T P C ( X ) a Again, the result follows from the fact that projection matrices are unique.
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This note was uploaded on 11/13/2011 for the course STAT 8260 taught by Professor Hall during the Summer '10 term at University of Georgia Athens.

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