Testing multinomial parameters under order restrictions

Let be a given probability vector .We consider the probability vector , defined by pi = λ iqi , for all i and satisfies an arbitrary quasi-order restriction. Given observations from a multinomial distribution with probability vector p, we consider estimation of p based on the directed divergence criteria subject to such restrictions. We consider the likelihood ratio tests when testing for and against the order restriction, and in both cases the asymptotic null distributions of the test statistics are shown to be of the chi-bar-squared type. When the directed divergence is used as a test criteria, the null asymptotic distribution of the test statistic is shown to be the same as the corresponding null asymptotic distribution of the likelihood ratio statistic in both cases. An example is used to illustrate the technique developed. A simulation is conducted to compare the performance of different test statistics.