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Unformatted text preview: Tutorial 9: solutions 1. Suppose we need to estimate a varying coefficient model Y = a ( x 1 ) + a 1 ( x 2 ) x 3 + ε with sample ( x i 1 , ..., x i 3 , Y i ) , i = 1 , ..., n . Using cubic spline to approximate a k ( z ). (a) write the expression for the estimator of a 1 ( z ) (b) find the 95% confidence band for a 1 ( x ). (a) Suppose we choose t 1 ,...,t J as knots for a ( . ) and thus we have approximately a ( x ) = J +4 X j =1 θ j B j ( x ) where base funcitons B 1 ( x ) = 1 ,B 2 ( x ) = x,..., B J +4 ( x ) = ( x t J ) 3 + . Suppose we choose t 1 ,...,t J 1 as knots for a 1 ( . ) and thus we have approximately a 1 ( x ) = J 1 +4 X j =1 γ j B j ( x ) where base funcitons B 1 ( x ) = 1 ,B 2 ( x ) = x,..., B J 1 +4 ( x ) = ( x t J 1 ) 3 + . The model is now Y = J +4 X j =1 θ j B j ( x ) + J 1 +4 X j =1 γ j { B j ( x ) x 1 } + ε or Y = β > W + ε where β = ( θ 1 ,...,θ J +4 ,γ 1 ,...,γ J 1 +4 ) and W = ( B 1 ( x 1 ) ,...,B J +4 ( x 1 ) ,B 1 ( x 2 ) x 3 ,...,B J +4 ( x 2 )...
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This note was uploaded on 10/04/2010 for the course STAT ST4240 taught by Professor Xiayingcun during the Fall '09 term at National University of Singapore.
 Fall '09
 XIAYingcun

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