In testing hypotheses involving order restrictions on a collection of parameters, distributions arise which are mixtures of standard distributions. Since tractable expressions for the mixing proportions generally do not exist even for parameter collections of moderate size, the implementation of these tests may be difficult. Stochastic upper and lower bounds are obtained for such test statistics in a variety of these kinds of problems. These bounds are also shown to be tight. The tightness results point out some situations in which the bounds could be used to obtain approximate methods. These results can also be applied to obtain the least favorable configuration when testing the equality of two multinomial populations versus a stochastic ordering alternative.
"Bounds on Mixtures of Distributions Arising in Order Restricted Inference." Ann. Statist. 10 (1) 302 - 306, March, 1982. https://doi.org/10.1214/aos/1176345713