Testing for and against an Order Restriction on Multinomial Parameters

Likelihood-ratio statistics are considered for testing a simple null hypothesis on a collection of multinomial parameters against an order-restricted alternative and for testing an order restriction against all alternatives. For the former test, the asymptotic distribution of the test statistic under the null hypothesis is a version of the chi-bar-squared distribution. This extends the work of Chacko (1966). For the latter test, homogeneity is, asymptotically, the least favorable configuration and under this hypothesis the asymptotic distribution of the test statistic has tail probabilities which are weighted averages of standard chi-squared tail probabilities.