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Week12Student2009

# Week12Student2009 - Week 12 Lecture 22 A group of students...

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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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Week12Student2009 - Week 12 Lecture 22 A group of students...

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