BOUNDS ON DISTRIBUTIONS ARISING IN ORDER RESTRICTED INFERENCES WITH RESTRICTED WEIGHTS

Level probabilities, which are central to the theory of testing order restricted hypotheses, are intractable in some applications. Thus, approximations for them and the associated distributions are of considerable importance. The level probabilities depend on weights which in some situations are monotonic. For the simple linear and simple tree orderings, we study the improvements in the equal-weights and pattern approximations which arise when the weights are monotonic. In this study, sharp upper and lower bounds for the related chi-bar-squared and E-bar-squared distributions are obtained for restricted weights. An application of these results in testing for a stochastic ordering between multinomial populations is discussed.