Page 237 - Elements of Distribution Theory

P1: JZX
052184472Xc07 CUNY148/Severini May 24, 2005 3:59

7.5 Order Statistics 223

Theorem 7.10. Let X 1 , X 2 ,..., X n denote independent, identically distributed, real-valued
random variables, each with an absolutely continuous distribution with density p and
distribution function F. Let X (1) , X (2) ,..., X (n) denote the order statistics of X 1 ,..., X n
and let m < r.
The distribution of (X (m) , X (r) ) is absolutely continuous with density function

n! m−1 r−m−1 n−r
F(x m ) [F(x r ) − F(x m )] [1 − F(x r )] p(x m )p(x r ),
(m − 1)!(r − m − 1)!(n −r)!
for x m < x r .

Proof. The density function of (X (1) , X (2) ,..., X (n) )isgiven by
n!p(x 1 ) ··· p(x n ), −∞ < x 1 < x 2 < ··· < x n < ∞.

The marginal density of (X (m) , X (r) ), m < r,is therefore given by
∞ ∞

n! ··· p(x 1 ) ··· p(x n )I {x 1 <x 2 <···<x n } dx 1 ··· dx m−1 dx m+1 ··· dx r−1 dx r+1 ··· dx n .
−∞ −∞
Note that

I {x 1 <x 2 <···<x n } = I {x 1 <···<x m } I {x m <x m+1 <···<x r } I {x r <x r+1 <···<x n } .
By Lemma 7.1,
∞ ∞ 1 m−1

··· p(x 1 ) ··· p(x m−1 )I {x 1 <x 2 <···<x m−1 <x m } dx 1 ··· dx m−1 = F(x m ) ,
(m − 1)!
−∞ −∞
∞ ∞

··· p(x m+1 ) ··· p(x r−1 )I {x m <x m+1 <···<x r−1 <x r } dx m+1 ··· dx r−1
−∞ −∞
1 r−m−1
= [F(x r ) − F(x m )] ,
(r − m − 1)!
and
∞ ∞

··· p(x r+1 ) ··· p(x n )I {x r <x r+1 <···<x n−1 <x n } dx r+1 ··· dx n
−∞ −∞
1 n−r
= [1 − F(x r )] .
(n − r)!
The result follows.

Example 7.25 (Order statistics of exponential random variables). Let X 1 , X 2 ,..., X n
denote independent, identically distributed random variables, each with an exponential
distribution with parameter λ> 0; this distribution has density function λ exp(−λx), x > 0,
and distribution function 1 − exp(−λx), x > 0.
According to Theorem 7.9, (X (1) ,..., X (n) ) has an absolutely continuous distribution
with density function

n

n
n!λ exp −λ x j , 0 < x 1 < x 2 < ··· < x n < ∞.
j=1