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Week9Student2009

# Week9Student2009 - Week 9 Lecture 16 Quadratic functional...

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Unformatted text preview: Week 9 Lecture 16 Quadratic functional estimation Model: Observe the sequence model: y i = & i + n & 1 = 2 z i where z i i:i:d: & N (0 ; 1) . The model comes from the white noise model (or many other models): dy ( t ) = f ( t ) dt + n & 1 = 2 dB ( t ) , t 2 [0 ; 1] . Let f ¡ i ( t ) ; i = 1 ; 2 ;::: g be an orthonormal basis of L 2 ([0 ; 1]) . The white noise problem is then equivalent to the sequence model with y i = h ¡ i ;y i ; & i = h ¡ i ;f i ; and z i = h ¡ i ;B i i:i:d: & N (0 ; 1) . Assumption : Let & = ( & 1 ;& 2 ;::: ) . Assume & &;M & = ( & : X i =1 i 2 & & 2 i ¡ M ) . If the orthonormal basis is the Fourier basis, this assumption corresponds to the periodic Sobolev ball with smoothness ¢ . Now we may write the sequence model as P n = f P n;¡ ; & 2 & &;M g where P n;¡ is the joint distribution of independent y i & N ( & i ; 1 =n ) . Problem : Estimate Q = P i =1 & 2 i ( or R f 2 with f = P i =1 & i ¡ i ), and determine the optimal minimax rate n & ¢ satisfying lim n...
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Week9Student2009 - Week 9 Lecture 16 Quadratic functional...

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