This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full
Document
 Fall '09
 XIAYingcun

Click to edit the document details