Order Statistics

Order Statistics - Order Statistics Let X1, X 2 , , X n be...

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Order Statistics Let 12 ,,, n XX X be a random sample from the density () X fx and suppose that we want the joint density of n YY Y where the i Y are the i X arranged in order of magnitude so that n Y ≤≤≤ . In what follows, we develop the expression for the joint density for 3 n = . We assume that X is the density of a continuous random variable. Let 123 ,, XXX be a sample of size 3 from X . We shall transform from the x’s to the y’s as follows Let 1. 11 22 33 ;; YX === if {} 1 1 2 3 ,, | I XXX X X X =< < 2. 23 32 if 2 1 3 2 I X < 3. 21 if 3 2 1 3 I XXX X X X < 4. 31 if 4 2 3 1 I X X < 5. 13 if 5 3 1 2 I < 6. if 6 3 2 1 I X X < Note that 0 jk j k II I I •= = (i. .e. the regions in X space are pair-wise disjoint) and 123456 1 2 3 1 2 3 , , IIIIII X X X X X X +++++= −∞ << ∞−∞ is the entire space. We shall find the joint density of YYY for each of the above six cases or regions. In all cases, the joint density of is ( ) ( ) ( ) 1 2 3 X X X f xxx fxfxfx = . Case 1 For the region 1 1 2 3 I X < , the Jacobian is one and the joint density is ( ) ( ) ( ) 1 2 3 X X X f yyy fyfyfy = for and zero elsewhere. Case 2 For the region 2 1 3 2 I X < , the Jacobian is one and the joint density is ( ) ( ) ( ) 1 3 2 X X X f = for and zero elsewhere.
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Case 3 For the region () {} 123 3 2 1 3 ,, ,, | XXX I XXX X X X =< < , the Jacobian is one and the joint density is ( ) ( ) ( ) 2 1 3 YYY X X X f yyy fyfyfy = for << and zero elsewhere. Case 4 For the region 4 2 3 1 I X X < , the Jacobian is one and the joint density is ( ) ( ) ( ) 2 3 1 X X X f = for and zero elsewhere. Case 5 For the region 5 3 1 2 I < , the Jacobian is one and the joint density is ( ) ( ) ( ) 3 1 2 X X X f = for and zero elsewhere.
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Order Statistics - Order Statistics Let X1, X 2 , , X n be...

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