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Week1Student2009

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

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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 ±² exp " ± ( x ± ° ) 2 2 ² 2 # ; ° 2 R; ² > 0 . 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 ² 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 ³ + ´ i , i = 1 ; 2 ; : : : ; n where p is ±xed, ³ = ° ³ 1 ; ³ 2 ; : : : ; ³ 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 ³ . Nonparametric regression : Y i = f ( X i ) + ´ i , i = 1 ; 2 ; : : : ; n For p = 1 , X i = i=n , we have Y i = f ² i n ³ + ´ i , i = 1 ; 2 ; : : : ; n We may assume that f 2 F = n g : R ( g ±( x )) 2 dx ² M o . Estimation problems: f , R f 2 (Bickel, et. al.), f ( x 0 ) (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

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Kernel method Histograms: de±ne bins B i = ´ i ± 1 m ; i m ³ , 1 ² i ² m = [( i ± 1) h; ih ) , h = 1 =m . Let N i be the number of observations in B i , and p i = Z B i f t f ( x ) h , x 2 B i then b p i = N i n leads b f n ( x ) = 1 n N i h , x 2 B i = 1 n n X i =1 1 h I (( i ± 1) h ² Y i < ih ) , x 2 B i A Variation of Histograms: b f n ( x ) = 1 n n X i =1 I ( x ± h= 2 ² Y i < x + h= 2) h = 1 n n X i =1 1 h I ² ± 1 = 2 ² Y i ± x h < 1 = 2 ³ because P n i =1 I ( x ± h= 2 ² Y i < x + h= 2) n !
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