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# DMtutorial7s - Tutorial 8 suggested solutions 1 suppose we...

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Tutorial 8: suggested solutions 1. suppose we have sample ( x i , y i ) , i = 1 , ..., n . we estimate the regression model y i = g ( x i ) + ε i , where g ( x ) is a spline function of the form g ( x ) = J +4 X j =1 β j B j ( x ) (a) Estimate the derivative g 0 ( x ) of g ( x ) (b) ﬁnd the 95% conﬁdence band for g 0 ( x ). Let X = B 1 ( x 1 ) B 2 ( x 1 ) ... B J +4 ( x 1 ) B 1 ( x 2 ) B 2 ( x 2 ) ... B J +4 ( x 2 ) ... B 1 ( x 1 ) B 2 ( x 1 ) ... B J +4 ( x 1 ) Y = Y 1 Y 2 ... Y n and ˆ β 1 ˆ β 2 ... ˆ β n = ( X > X ) - 1 X > Y The estimated funciton is ˆ g ( x ) = J +4 X j =1 ˆ β j B j ( x ) (a) g 0 ( x ) can be estimated by ˆ g 0 ( x ) = J +4 X j =1 ˆ β j B 0 j ( x ) = J +4 X j =2 ˆ β j B 0 j ( x ) where B 1 ( x ) = 1 , B 2 ( x ) = x, B 3 ( x ) = x 2 , B 4 ( x ) = x 3 , B 0 j +4 ( x ) = ( x - t j ) 3 + , j = 1 , ..., J thus B 0 2 ( x ) = 1 , B 0 3 ( x ) = 2 x, B 0 4 ( x ) = 3 x 2 , B 0 j +4 ( x

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DMtutorial7s - Tutorial 8 suggested solutions 1 suppose we...

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