6. Let X and Y be independently distributed according to one-parameter exponential families, so that their joint distribution is given by dP8! . 82 ( x, y) = C( °1) e 8•T(x) d",( x) K( 02) e 82U( y ) dJl(y ). Suppose that with probability 1 the statistics T and U each take on at least three values and that

(a,

b) is an interior point of the natural parameter space. Then a UMP unbiased test does not exist for testing H : 01 =

a, O2 = b against the alternatives 9. oF a or 92 oF b.* [The most powerful unbiased tests against the alternatives °1 "*

a, 02 = b and 01 =

a, O2 "* b have acceptance regions C1 < T(x) < C2 and K1 < U( y) < K2 respectively. These tests are also unbiased against the wider class of alternatives K: °1 "* a or 02 "* b or both.]