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1 S T 5 2 7 N o n p a r a m e t r i c R e g r e s s i o n , L e c t u r e 9 L i j i a n Y a n g D e p a r t m e n t o f S t a t i s t i c s & P r o b a b i l i t y M i c h i g a n S t a t e U n i v e r s i t y E a s t L a n s i n g , M I 4 8 8 2 4 a n d D e p a r t m e n t o f S t a t i s t i c s & A p p l i e d P r o b a b i l i t y N a t i o n a l U n i v e r s i t y o f S i n g a p o r e S i n g a p o r e 1 1 7 5 4 6 S T 5 2 7 N o n p a r a m e t r i c R e g r e s s i o n , 2 4 t h M a r c h 2 5 2 S p l i n e e s t i m a t i o n m p ( x ) = e m p ( x ) + e p ( x ) w i t h b o t h t e r m s a s { b 1- p , p ( x ) , . . . , b N , p ( x ) } n ( h b j , p , b j , p i n ) N j , j = 1- p o- 1 h b j , p , m i n N j = 1- p { b 1- p , p ( x ) , . . . , b N , p ( x ) } n ( h b j , p , b j , p i n ) N j , j = 1- p o- 1 h b j , p , E i n N j = 1- p e m p ( x ) i s t h e m i n i m i z e r o f k g- m k 2 2 , n k e m p- m k 2 , n k g p- m k 2 , n k g p- m k C p m ( p ) h p k e m p- g p k 2 , n 2 C p m ( p ) h p k m p- m k 2 , n k e m p- m k 2 , n + k e p k 2 , n , fi r s t t e r m i s b o u n d e d b y h p , o n l y n e e d s t o b o u n d t h e s e c o n d t e r m W h a t a b o u t k e p k 2 , n ? S T 5 2 7 N o n p a r a m e t r i c R e g r e s s i o n , 2 4 t h M a r c h 2 5 3 A s y m p t o t i c r e s u l t s W i t h E ( X i ) = ( X i ) i , i = 1 , . . . , n , t h e n o i s e t e r m p ( x ) i s { b 1- p , p ( x ) , . . . , b N , p ( x ) } n ( h b j , p , b j , p i n ) N j , j = 1- p o- 1 h b j , p , E i n N j = 1- p F o r p = 1 , 2 , w e c a n s h o w t h a t a s n s u p g 1 , g 2 G ( p- 2 ) fl fl fl fl h g 1 , g 2 i n- h g 1 , g 2 i k g 1 k 2 k g 2 k 2 fl fl fl fl = O p r N l o g ( n ) n !... View Full Document

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