9 - NOTES 9 Forecasting: Mean squares error prediction Give...

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NOTES 9 Forecasting : Mean squares error prediction Give a random vector ( Y, X ) 0 , where Y is a scalar and X is a vector. We want to get a function g ( X ) which predict Y as closely as possible, in the Mean squares error sense, i.e MSE = E [ Y - g ( X )] 2 is to be minimized. Claim : g ( X ) = E ( Y | X ) is the optimal choice. Here E ( Y | X ) is called the Mean squares error predictor. Proof of the Claim : First observe that E [( Y - c ) 2 ] = E [( Y - μ ) + ( μ - c )] 2 , μ = E ( Y ) = E ( Y - μ ) 2 + ( μ - c ) 2 + 2( μ - c ) E ( Y - μ ) = E ( Y - μ ) 2 + ( μ - c ) 2 E ( Y - μ ) 2 with equality holds if and only if c = μ = E ( Y ). That is, E [( Y - c ) 2 ] is minimized at c = E ( Y ). Recall that E [( Y - c ) 2 ] = E [( Y - E ( Y )) 2 ] + [ E ( Y ) - c ] 2 . For any function g ( X ), put c = g ( x ). Take expectation of Y given X = x . Then we have E [( Y - g ( x )) 2 | X = x ] = E [( Y - E ( Y | X = x )) 2 | X = x ] +[ E ( Y | X = x ) - g ( x )] 2 Therefore E [( Y - g ( X )) 2 | X ] = E [( Y - E ( Y
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This note was uploaded on 11/02/2011 for the course STAT 4005 taught by Professor Wu,kaho during the Spring '08 term at CUHK.

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9 - NOTES 9 Forecasting: Mean squares error prediction Give...

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