Page 239 - Elements of Distribution Theory

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

7.6 Ranks 225

that X 1 ,..., X n are unique with probability 1. Let R = (R 1 ,..., R n ) denote the vector of
ranks.
The following theorem summarizes the properties of R.

Theorem 7.11. Let X 1 ,..., X n denote independent, identically distributed, real-valued
random variables, each with an absolutely continuous distribution. Then
(i) The statistic (R, X (·) ) is a one-to-one function of X.
(ii) (R 1 ,..., R n ) is uniformly distributed on the set of all permutations of (1, 2,..., n);
that is, each possible value of (R 1 ,..., R n ) has the same probability.
(iii) X (·) and R are independent
(iv) For any statistic T ≡ T (X 1 ,..., X n ) such that E(|T |) < ∞,

E[T |R = r] = E[T (X (r 1 ) , X (r 2 ) ,..., X (r n ) )]
where r = (r 1 ,r 2 ,...,r n ).

Proof. Clearly, (R, X (·) )isa function of (X 1 ,..., X n ). Part (i) of the theorem now follows
from the fact that X j = X (R j ) , j = 1,..., n.
Let τ denote a permutation of (1, 2,..., n) and let
n
X(τ) ={x ∈ X : x τ 1 < x τ 2 < ··· < x τ n }.
Let
X 0 =∪ τ X(τ)

where the union is over all possible permutations of (1, 2,..., n). Note that
Pr{(X 1 ,..., X n ) ∈ X 0 }= 1

so that we may proceed as if the range of (X 1 ,..., X n )is X 0 .
Let h denote a real-valued function of R ≡ R(X) such that E[h(R)] < ∞. Then

E[h(R)] = E[h(R(X))I {X∈X(τ)} ].
τ
Note that, for X ∈ X(τ), R(X) = τ. Hence,

E[h(R)] = E[h(τ)I {X∈X(τ)} ] = h(τ)Pr(X ∈ X(τ)).
τ τ
Let τ 0 denote the identity permutation. Then
X ∈ X(τ) if and only if τ X ∈ X(τ 0 )

and,sincethedistributionof(X 1 , X 2 ,..., X n )isexchangeable,τ X hasthesamedistribution
as X. Hence,

Pr(X ∈ X(τ)) = Pr(τ X ∈ X(τ 0 )) = Pr(X ∈ X(τ 0 )).
Since there are n! possible permutations of (1, 2,..., n) and

Pr(X ∈ X(τ)) = 1,
τ