This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: February 19 th Order Statistics Let 1 2 , , , n X X X L be a random sample from a population with p.d.f. ( ) f x . Then, (1) 1 2 ( ) 1 2 min( , , , ) max( , , , ) n n n X X X X X X X X = = L L and (1) (2) ( ) ( ) k n X X X X L L p.d.f.s for (1) ( ) and n X X W.L.O.G.(Without Loss of Generality), lets assume X is continuous. (1) 1 2 ( ) ( , , , ) n P X x P X x X x X x = f f f L f 1 1 ( ) [1 ( )] i n n i X i i P X x F x = = = = f (1) (1) 1 1 1 1 ( ) 1 [1 ( )] ( ) [1 ( )] [1 ( )] [1 ( )] ( ) i i n X X i n n n n X X i i F x F x d d f x F x F x n F x f x dx dx = = = =  =  =  =  ( ) ( ) 1 2 ( ) ( ) ( , , , ) n X n n F x P X x P X x X x X x = = L 1 ( ) [ ( )] i n n X i F x F x = = = ( ) 1 ( ) [ ( )] ( ) n n X f x n F x f x = Example. Let . . . ~ exp( ). 1, , i i d i X i n = L Please 1. Derive the MLE of 2. Derive the p.d.f. of (1) X 3. Derive the p.d.f. of ( ) n X Solutions. 1. 1 1 1 ( ) ( ) n i i i n n x x n i i i L f x e e = = = = = = 1 ln ln n i i l L n x = = = 1 0 i dl n x d X = = = Is an unbiased estimator of ? 1 ? E X = ( ) i X M t t = ( ) i n X M t t =  ~ ( , ) i Y X gamma n = ( 29 1 ( ) ( ) i n y Y X e y f x n  = = Let 1 n i i Y X = = ( 29 1 1 1 ( 1)!...
View
Full
Document
This note was uploaded on 11/14/2010 for the course AMS 312 taught by Professor Zhu,w during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Zhu,W

Click to edit the document details