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

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

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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

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