Page 235 - Elements of Distribution Theory

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

7.5 Order Statistics 221

Proof. Fix s, t;if t ≤ s, then X (r) ≤ t implies that X (m) ≤ t ≤ s, m < r,so that

Pr(X (m) ≤ s, X (r) ≤ t) = Pr(X (r) ≤ t).
Now suppose that s < t; let N 1 denote the number of the observations X 1 ,..., X n falling
in the interval (−∞, s], let N 2 denote the number of observations falling in the interval (s, t],
and let N 3 = n − N 1 − N 2 . Then, for m < r,
Pr(X (m) ≤ s, X (r) ≤ t) = Pr(N 1 ≥ m, N 1 + N 2 ≥ r)
n n−i

= Pr(N 1 = i, N 2 = j).
i=m j=max(0,r−i)
Since (N 1 , N 2 , N 3 ) has a multinomial distribution, with probabilities F(s), F(t) −
F(s), 1 − F(t), respectively, it follows that

Pr(X (m) ≤ s, X (r) ≤ t)
n n−i
n i j n−i− j
= F(s) [F(t) − F(s)] [1 − F(t)] ,
i, j, n − i − j
i=m j=max(0,r−i)
as stated in part (i).
If the distribution function F in Theorem 7.8 is absolutely continuous, then the dis-
tribution of the order statistics (X (m) , X (r) )is absolutely continuous and the corresponding
density function may be obtained by differentiation, as in Theorem 7.7. However, somewhat
suprisingly, it turns out to be simpler to determine the density function of the entire set of
order statistics and then marginalize to determine the density of the pair of order statistics
under consideration.

Theorem 7.9. Let X 1 , X 2 ,..., X n denote independent, identically distributed real-valued
random variables each with distribution function F. Suppose that the distribution function F
isabsolutelycontinuouswithdensity p.Thenthedistributionof(X (1) ,..., X (n) )isabsolutely
continuous with density function
n!p(x 1 ) ··· p(x n ), x 1 < x 2 < ··· < x n .

Proof. Let τ denote a permutation of the integers (1, ··· , n) and let

n
X(τ) ={x ∈ X : x τ 1 < x τ 2 < ··· < x τ n }
where X denotes the range of X 1 . Let
X 0 =∪ τ X(τ)

where the union is over all possible permutations; note that
Pr{(X 1 ,..., X n ) ∈ X 0 }= 1

and, hence, we may proceed as if X 0 is the range of X = (X 1 ,..., X n ).
), let X (·) = (X (1) ,..., X (n) ) denote the vector of order statistics,
Let τ X = (X τ 1 ,..., X τ n
and let h denote a bounded, real-valued function on the range of X (·) . Then

E[h(X (·) )] = E{h(X (·) )I {X∈X(τ)} }.
τ