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Unformatted text preview: Week 13 Lecture 24 Adaptive Wavelet Estimation Donoho and Johnstone (1995, JASA). Sketch of the proof. Consider the sequence model where yi = i + zi , i = 1; :::; d and zi are independent normal N (0; 1) variables. Set r( ) = d 1 Pk^ k2 . The stein’ s 2 unbiased estimator of risk gives r( ; L) = P (U ( )) with U( ) =
d P i=1 1 + 2I (jyi j 2 # fi; jyi j 2 ) + yi I (jyi j )+
2 2 =d When y(i) g+ < y(i+1) , U ( ) is an increasing function of n. This implies
s P I (jyi j > ) (jyi j ^ ) = arg min U ( ) 2 f0; jy1 j ; : : : ; jyn jg . Proposition: P
0 sup p 2 log d jU ( ) r ( ; L)j C log3=2 d : d 1 =2 Remark: This proposition inspires them to construct an estimator through 3 P2 1 2 a hybrid method. Set Td = d 1 xi 1 , d = d 2 log2 d and De…ne the estimator b(x) of by and Proof: Set It is easy to see jYi ( )j P jZd ( )j b(x) = ^ b =^ i
2
S ; if Td > d, F ; if Td r( ) = d: Zd ( ) = Ud ( ) 2+ . Then Hoe¤ding inequality implies ! ! C log3=2 d C 2 log3 d 2 exp : 2 d 1=2 2 2+ 2 F P Yi ( ) . 1 De…ne ( j =j d. Note that ) 3C log3=2 d sup jZd ( )j p d 1=2 0 2 log d ( )( C log3=2 d sup jZd ( )j [ sup sup d 1=2 j jt tj j j d jZd (t) Zd (tj )j 2C log3=2 d d 1=2 ) It is easy to see sup
jt tj j
d jZd (t) Zd (tj )j 2d
1=2 1 (1 + d td ) # fi; jti tj j dg + 5 d td where d td is chosen to be o d . More details will be given in class. 2 Lecture 25 High dimensional estimation? Beyond wavelets? ... 3 ...
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 '09
 HarrisonH.Zhou
 Trigraph, zd, sup sup, Wavelet Estimation Donoho

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