Week2Student2009

Week2Student2009 - Week 2 Lecture 3 General m Model: Let Y...

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Unformatted text preview: Week 2 Lecture 3 General m Model: Let Y 1 ; Y 2 , . .., Y n i.i.d. on [0 ; 1] with density f 2 F , F = n f; R & f ( m ) ( x ) ¡ 2 dx & M o Goal: Find b f such that sup f 2F E Z ¢ b f ( x ) ¡ f £ 2 & C M n & 2 m= (2 m +1) Find a Kernel K such that Z K ( y ) dy = 1 Z y k K ( y ) dy = ; k = 1 ;:::;m ¡ 1 Z j y j m j K ( y ) j dy & M 1 Z K 2 ( y ) dy & M 2 (Note that K may not be nonnegative). The bias part is E b f n ( x ) ¡ f ( x ) = Z K ( y ) [ f ( x + hy ) ¡ f ( x )] dy = Z K ( y ) " m & 1 X k =1 f ( k ) ( x ) ( hy ) k k ! + ( hy ) m Z 1 f ( m ) ( x ¡ shy ) (1 ¡ s ) m & 1 ( m ¡ 1)! ds # dy = Z Z 1 K ( y ) ( hy ) m f ( m ) ( x ¡ shy ) (1 ¡ s ) m & 1 dsdy The Cauchy-Schwartz inequality implies ¤Z Z 1 K ( y ) y m f ( m ) ( x ¡ shy ) (1 ¡ s ) m & 1 dsdy ¥ 2 & ¤Z Z 1 j K ( y ) j 1 = 2 j y j m= 2 (1 ¡ s ) ( m & 1) = 2 ¢ j K ( y ) j 1 = 2 j y j m= 2 f ( m ) ( x ¡ shy ) (1 ¡ s ) ( m & 1) = 2 dsdy ¥ 2 & Z Z 1 j K ( y ) jj y j m (1 ¡ s ) m & 1 dsdy Z Z 1 j K ( y ) jj y j m ¢ f ( m ) ( x ¡ shy ) £ 2 (1 ¡ s ) m & 1 dsdy . Then Z ¢ E b f n ( x ) ¡ f ( x ) £ 2 dx & h 2 m M " Z Z 1 j K ( y ) jj y j m (1 ¡ s ) m & 1 ( m ¡ 1)! dsdy # 2 & h 2 m MM 1 = ( m !) 2 . 1 Now we consider the variance var & b f n ( x ) ¡ = var 1 n n X i =1 1 h K ¢ Y i & x h £ ! = 1 n var ¢ 1 h K ¢ Y i & x h ££ ¡ 1 nh 2 E ¤ K ¢ Y i & x h £¥ 2 = 1 nh Z K 2 ( y ) f ( x + hy ) This implies Z var & b f n ( x ) ¡ ¡ M 2 1 nh This gives Z E & b f n ( x ) & f ( x ) ¡ 2 ¡ h 2 m MM 2 1 = ( m !) 2 + M 2 1 nh If you choose h such that h 2 m MM 2 1 = ( m !) 2 = M 2 1 nh we then have Z E & b f n ( x ) & f ( x ) ¡ 2 ¡ C M;K n & 2 m= (2 m +1) . Questions: ¢ The existence of such a K ? For instance, we could &nd K with support in [ & 1 ; 1] . The reason is as follows. Consider L 2 ([ & 1 ; 1]) , and a subspace S generated by ¦ y;y 2 ;:::;y...
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This note was uploaded on 11/06/2009 for the course STAT 680 at Yale.

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Week2Student2009 - Week 2 Lecture 3 General m Model: Let Y...

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