Week4Student2009

Week4Student2009 - Week 4 Lecture 7 Optimal rate of convergence in the sup-norm Model Let Y 1 Y 2 Y n i.i.d on[0 1 with density f 2 F& M

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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Week 4 Lecture 7 Optimal rate of convergence in the sup-norm Model : Let Y 1 ; Y 2 , . . . , Y n i.i.d. on [0 ; 1] with density f 2 F & ( M ) , H&lder ball of order & . Minimax rate : It can be shown that inf ^ f sup F & ( M ) E & & & b f & f & & & 2 1 ¡ C ¡ n log n ¢ & 2 &= (2 & +1) . For simplicity we will only show the case & = 1 to reveal the ¡mysterious¢ log term. Upper bound : In this lecture we will show there is a estimator b f such that sup F & ( M ) E & & & b f & f & & & 2 1 ¢ C ¡ n log n ¢ & 2 &= (2 & +1) where & = 1 . A histogram estimator: Let r = b 1 =h c . De£ne b f ( x ) = 1 n n X i =1 I (( r & 1) h ¢ Y i < rh ) h , ( r & 1) h ¢ x < rh . Denote Z i = 1 h I (( r & 1) h ¢ Y i < rh ) & f ( x ) = EZ i = 1 h Z rh ( r & 1) h f ( x ) dx , for ( r & 1) h ¢ x < rh Bias: sup x £ £ f ( x ) & & f ( x ) £ £ 2 = sup r £ £ £ £ £ f ( x ) & 1 h Z rh ( r & 1) h f ( x ) dx £ £ £ £ £ 2 f ( r & 1) h ¢ x < rh g ¢ Ch 2 ....
View Full Document

This note was uploaded on 11/06/2009 for the course STAT 680 at Yale.

Page1 / 5

Week4Student2009 - Week 4 Lecture 7 Optimal rate of convergence in the sup-norm Model Let Y 1 Y 2 Y n i.i.d on[0 1 with density f 2 F& M

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online