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Unformatted text preview: Week 12 Lecture 22 A group of students present Donoho and Johnstone (PTRF, 1994)? 1 Lecture 23 Review from the presentation Suppose we observe y i = & i + z i ; i = 1 ; :::; n , = 1 where & is constrained to lie in a ball of radius C de&ned by l p norm, & = & p;c = n &; k & k p & C o = & &; 1 n k & k p p & 1 n C p . Let p = 1 n C p . We want to evaluate R N ( & ) = inf b & sup & P b & & 2 2 . It can be shown R N ( & ) sup P & 2M B ( P & ) = nB ( P & 1 ) where B ( P & ) = inf b & P & P y j & & b & 2 , P = & P & : 1 n X P & j & i j p & p and B ( P & 1 ) = inf b & 1 P & 1 P y 1 j & 1 & 1 b & 1 2 , P 1 = f P & 1 : P & 1 j & 1 j p & p g . Theorem Let y = & + z; z N (0 ; 1) , and P = f P & : P & j & j p & p g . As ! ; B ( ) & 2 2 & p & 1 p (2 log & p ) 1 & p= 2 < p < 2 If p 2 , the is asymptotically minimax and P & = ( + & ) = 2 is asymptot ically least favorable. If < p < 2 , then b & with threshold...
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This note was uploaded on 11/06/2009 for the course STAT 680 at Yale.
 '09
 HarrisonH.Zhou

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