6.11 Prove the following generalization of Theorem 6.12 to families without common support.

Theorem 9.1 Let P be a finite family with densities pi, i = 0,...,k, and for any x, let S(x) be the set of pairs of subscripts (i, j ) for which pi(x) + pj (x) > 0. Then, the statistic T (X) = pj (X)

pi(X)

, i < j and (i, j ) ∈ S(X)

"

is minimal sufficient. Here, pj (x)/pi(x) = ∞ if pi(x)=0 and pj (x) > 0.