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Download Ordered Random Variables: Theory and Applications by Muhammad Qaiser Shahbaz, Mohammad Ahsanullah, Saman Hanif PDF

By Muhammad Qaiser Shahbaz, Mohammad Ahsanullah, Saman Hanif Shahbaz, Bander M. Al-Zahrani

Ordered Random Variables have attracted numerous authors. the elemental development block of Ordered Random Variables is Order facts which has numerous purposes in severe price conception and ordered estimation. the overall version for ordered random variables, referred to as Generalized Order records has been brought rather lately via Kamps (1995).

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58) as p,q p,q μr,s:n − μr,s−1:n = q Cr,s:n n−s+1 ∞ −∞ ∞ x1 p q−1 x1 x2 × [F(x1 )]r −1 F(x2 ) − F(x1 ) × [1 − F(x2 )]n−s+1 d x 2 d x1 .

Now the conditional distribution of X r :n given X s:n = xs is f (xr |xs ) = = fr,s:n (xr , xs ) f s:n (xs ) n! (n − s)! r −1 × [F(xs ) − F(xr )]s−r −1 [1 − F(xs )]n−s / n! (n − s)! or f (xr |xs ) = (s − 1)! (s − r − 1)! s−r −1 × F(xs ) − F(xr ) = 1 [F(xs )]s−1 f (xr ) F(xr ) (s − 1)! (s − r − 1)! F(xs ) F(xs ) × 1− F(xr ) F(xs ) r −1 r −1 s−r −1 . 21) 22 2 Order Statistics Proof immediately follows by noting that f (xr )/F(xs ) and F(xr )/F(xs ) are respectively the density and distribution function of a random variable whose distribution is truncated at right of xs .

2008). We know that when we have a bivariate distribution; say f (x, y); of two random variables X and Y , then the conditional distribution of random variable Y given X is given as f (x, y) ; f (y|x) = f 2 (x) 20 2 Order Statistics where f 2 (x) is the marginal distribution of X . Analogously, the conditional distributions in case of order statistics can be easily defined; say for example the conditional distribution of X s:n given X r :n = xr is defined as f (xs |xr ) = fr,s:n (xr , xs ) , fr :n (xr ) where fr,s:n (xr , xs ) is joint distribution of X r :n and X s:n and fr :n (x) is marginal distribution of X r :n .

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