Page 240 - Elements of Distribution Theory

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

226 Distribution Theory for Functions of Random Variables

it follows that Pr(X ∈ X(τ)) = 1/n! for each τ and, hence, that
1
E[h(R)] = h(τ)
n!
τ
proving part (ii).
Part (iii) follows along similar lines. Let g denote a bounded function of X (·) and let h
denote a bounded function of R.
Note that

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

= E[g(τ X)h(τ)I {X∈X(τ)} ]
τ

= h(τ)E[g(τ X)I {τ X∈X(τ 0 )} ]
τ

= h(τ)E[g(X)I {X∈X(τ 0 )} ]
τ

= E[g(X)I {X∈X(τ 0 )} ] h(τ)
τ
1
= n!E[g(X)I {X∈X(τ 0 )} ] h(τ)
n!
τ
= E[g(X (·) )]E[h(R)],

proving part (iii).
Finally, part (iv) follows from the fact that any statistic T ≡ T (X) may be written as
¯
T (R, X (·) ) and, by part (iii) of the theorem,
¯
¯
E[T |R = r] = E[T (R, X (·) )|R = r] = E[T (r, X (·) )] = E[T (X (r 1 ) , ··· , X (r n ) )].
Example 7.27 (Mean and variance of linear rank statistics). Let R 1 , R 2 ,..., R n denote
the ranks of a sequence of independent and identically distributed real-valued random
variables, each distributed according to an absolutely continuous distribution. Consider a
statistic of the form
n

T = a j R j
j=1
where a 1 , a 2 ,..., a n is a sequence of constants. Here we consider determination of the
mean and variance of T .
Note that each R j has the same marginal distribution. Since

n
n(n + 1)
R j = , (7.2)
2
j=1
it follows that E(R j ) = (n + 1)/2, j = 1,..., n. Also, each pair (R i , R j ) has the same
marginal distribution so that Cov(R i , R j ) does not depend on the pair (i, j).
2
Let σ = Var(R j ) and c = Cov(R i , R j ). By (7.2),
n

2
Var R j = nσ + n(n − 1)c = 0
j=1