Week1Student2009

Week1Student2009 - Week 1 Lecture 1 &An overview...

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Unformatted text preview: Week 1 Lecture 1 &An overview Introduction: Parametric Estimation vs. Nonparametric Estima- tion I: Parametric density estimation : Let Y 1 , Y 2 , ..., Y n i.i.d. with density f & ( x ) , & 2 & & R (or R 2 , or R 10 ). For instance, f &;¡ ( x ) = 1 p 2 ¡¢ exp " ¡ ( x ¡ & ) 2 2 ¢ 2 # ; & 2 R;¢ > . Nonparametric density estimation : Let Y 1 ; Y 2 , ..., Y n i.i.d. on [0 ; 1] with density f , f 2 F = f g : sup x j g ¡( x ) j ¢ M g (or balls in other function spaces & Sobolev spaces, Besov spaces, etc.) II: Parametric regression and Nonparametric regression: Linear regression Y i = X T i £ + ¤ i , i = 1 ; 2 ;:::;n where p is ¡xed, £ = & £ 1 ;£ 2 ;:::;£ p ¡ and X i = ( X i 1 ;X i 2 ;:::;X ip ) 2 R p . We may write Y i = f ( X i ) + ¤ i , i = 1 ; 2 ;:::;n with f ( X i ) = X T i £ . Nonparametric regression : Y i = f ( X i ) + ¤ i , i = 1 ; 2 ;:::;n For p = 1 , X i = i=n , we have Y i = f ¢ i n £ + ¤ i , i = 1 ; 2 ;:::;n We may assume that f 2 F = n g : R ( g ¡( x )) 2 dx ¢ M o . Estimation problems: f , R f 2 (Bickel, et. al.), f ( x ) (Farrell, et. al.), f ¡ , etc. De&nition Parametric estimation: observe Y with density f ( x;& ) , & 2 & & R k , k ¡xed constant. Nonparametric estimation: observe Y with density f ( x ) , f 2 F , a in¡nite dimensional function space. Some history : References: Rosenblatt (1956, AOS), Whittle (1958,JRSSB), Parzen (1962, AOS). 1 Kernel method Histograms: de&ne bins B i = & i & 1 m ; i m ¡ , 1 ¡ i ¡ m = [( i & 1) h;ih ) , h = 1 =m ....
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This note was uploaded on 11/06/2009 for the course STAT 680 taught by Professor Harrisonh.zhou during the Fall '09 term at Yale.

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Week1Student2009 - Week 1 Lecture 1 &An overview...

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