Bounds on Mixtures of Distributions Arising in Order Restricted Inference; (Distributional Bounds)

摘要

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 do not, in general, exist even for parameter collections of moderate size, the implementation of these tests may be difficult. It is not necessary to bound the individual mixing proportions in order to obtain a bound for the tail probabilities of the 'mixed' distributions. Bounds for such tail probabilities are derived and their application to several order restricted hypothesis testing situations is discussed. These results are also applied to obtain the least favorable configuration in a two sample likelihood ratio test of the equality of two multinomial populations versus a stochastic ordering alternative. (Author)