Page 236 - Elements of Distribution Theory

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222 Distribution Theory for Functions of Random Variables

Note that, for X ∈ X(τ), X (·) = τ X. Hence,

E{h(X (·) )I {X∈X(τ)} }= E{h(τ X)I {X∈X(τ)} }.
Let τ 0 denote the identity permutation. Then the event that X ∈ X(τ)is equivalent to the
event that τ X ∈ X(τ 0 ). Hence,

E[h(X (·) )] = E{h(τ X)I {τ X∈X(τ 0 )} }.
τ
Since X 1 ,..., X n are independent and identically distributed, for any permutation τ,
τ X has the same distribution as X.It follows that

E[h(X (·) )] = E{h(X)I {X∈X(τ 0 )} }= n!E[h(X)I {X∈X(τ 0 )} ];
τ
the factor n!is due to the fact that there are n! possible permutations. The result follows.

As noted above, the density function of (X (1) ,..., X (n) ) may be used to determine the
density function of some smaller set of order statistics. The following lemma is useful in
carrying out that approach.

Lemma 7.1. Let p denote the density function of an absolutely continuous distribution on
R and let F denote the corresponding distribution function. Then, for any n = 1, 2,...,
and any a < b,
∞ ∞

n
n! ··· p(x 1 ) ··· p(x n )I {a<x 1 <x 2 <···<x n <b} dx 1 ··· dx n = [F(b) − F(a)] .
−∞ −∞
Proof. Let X 1 , X 2 ,..., X n denote independent, identically distributed random variables,
each distributed according to the distribution with distribution function F and density
function p. Then, according to Theorem 7.9, the density function of (X (1) ,..., X (n) )is
given by

n!p(x 1 ) ··· p(x n ), −∞ < x 1 < x 2 < ··· < x n < ∞.
It follows that
Pr(a < X (1) < ··· < X (n) < b)
∞ ∞

= n! ··· p(x 1 ) ··· p(x n )I {a<x 1 <x 2 <···<x n <b} dx 1 ··· dx n .
−∞ −∞
Note that the event a < X (1) < ··· < X (n) < b is simply the event that all observations fall
in the interval (a, b). Hence,

Pr(a < X (1) < ··· < X (n) < b) = Pr(a < X 1 < b, a < X 2 < b,..., a < X n < b)
= Pr(a < X 1 < b) ··· Pr(a < X n < b)
n
= [F(b) − F(a)] ,
proving the result.

Using Lemma 7.1 together with Theorem 7.9 yields the density function of any pair of
order statistics; note that the same approach may be used to determine the density function
of any subset of the set of all order statistics.