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Unformatted text preview: Suggested Solutions for Tutorial 3 1. Suppose that X follows a uniform distribution, then for the inner points, the local linear kernel estimator and local constant (NW) estimator have the same asymptotic MSE. The MSE for NW kernel estimator is MSE NW = { 1 2 c 2 m 00 ( x ) h 2 + c 2 f 1 ( x ) f ( x ) m ( x ) h 2 } 2 + σ 2 d nhf ( x ) where c 2 = R K ( v ) v 2 dv and d = R K 2 ( v ) dv . The MSE for the local linear kernel estimator is MSE LL = { 1 2 c 2 m 00 ( x ) h 2 } 2 + σ 2 d nhf ( x ) . If X is uniformly distributed, then f ( x ) = 0. It follows MSE LL = MSE NW 2. For model Y = sin(2 πX ) + ε where X ∼ uniform (0 , 1) and ε is independent of X . If we apply local linear kernel estimation to estimate the curve, which point are expected to have the biggest absolute bias? The bias for the local linear kernel smoother is 1 2 c 2 m 00 ( x ) h 2 because sin 00 (2 πx ) = (2 π ) 2 sin (2 πx ), which has the largest absolute value as x = 1 / 4 and 3 / 4. Therefore the estimator has biggest bias at x = 1 / 4 and 3 / 4. 3. The uniform kernel function is defined as K ( v ) = 1 , if  v  < = 1 , , if  v  > 1 ....
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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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