57
Proof :
First, it will be shown that conditioning on the n c
type 2 and 3 pairs does not affect the exchangeability
property. Define W^, GW^ £ ^ and I(W^ < t) as in Lemma
3.3.1. Now under HQ for the entire sample, we have
(^1 i,X2i ^i) = (^2iXli>^i^
and applying an appropriate function (and Theorem 1.3.7 of
Randles and Wolfe, 1979)
d *
I(Wt < t)I(5ie(2,3)) = I(W. < t)I(f(5i)e(2,3)).
Taking expectations, it follows that
E{l(Wi Cj.Cj < c)
= Cj^lj ^ ^ c) = P ( Y j = 0>Cj c)
Noting that,
P(Yj = l.Cj < c) + P(Yj = O.Cj < c) = P(Cj < c)