Lecture6 - February 19 th Order Statistics Let 1 2 , , , n...

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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)!...
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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.

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Lecture6 - February 19 th Order Statistics Let 1 2 , , , n...

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