Summary: In testing hypotheses involving order restrictions on a collection of normal means, distributions arise which are mixtures of chi-square or beta distributions. In general the mixing coefficients are quite intractable even for a moderate number of populations.
Stochastic lower and upper bounds are obtained for the mixtures which arise in Bartholomew’s test for homogeneity of normal means with the alternative restricted by a partial ordering. The bounds are shown to be sharp. The techniques used also apply in the dual-testing situation, that is in testing the order restriction as the null hypothesis. The bounds also can be applied to obtain the least favorable distribution for testing the equality of two multinomial populations with a stochastically ordered alternative.