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

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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), let’s 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 λ = L Please 1. Derive the MLE of λ 2. Derive the p.d.f. of (1) X
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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 λ λ λ λ = - - = = = = = 1 ln ln n i i l L n x λ λ = = = - 1 ˆ 0 i dl n x d X λ λ λ = - = = Is ˆ λ an unbiased estimator of λ ? 1 ? E X = ( ) i X M t t λ λ = - ( ) i n X M t t λ λ = - ~ ( , ) i Y X gamma n λ = ( 29 1 ( ) ( ) i n y Y X e y f x n λ λ λ - - = = Γ Let 1 n i i Y X = = ( 29 1 0 1 1 ( 1)! n y e y E dy Y y n λ λ λ - - = - ( 29 2 0 1 ( 2)! n y e y dy n n λ λ λ λ - - = - -
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1 n λ = - 1 1 1 n E n X n n λ λ λ = = - - ˆ is not unbiased. λ 2.
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