iA ON LIKELIHOOD RATIO TESTS OF TREND

Suppose we have a random sample from a multinomial distribution with K parameters PI' P2' ••• 'PK ( ,r Pi=1) • Three likelihood ratio tests about 1=1 these parameters are considered: (i) -1 Test PI =P2 = ... =PK =K against an alternative specifying an order restriction on the p's • (ii) Test that the p's satisfy an order restriction against all alt~natives and (iii) Test PI ~ Pz ~ .•. ~ PK against PI < Pz ~ P3 ~ ••• ~ PK • For test (i) the asymptotic distribution under the null hypothesis is shown to be the x2 first studied by Bartholomew. For tests (ii) and (iii) homogeneity is found to be an asymptotically least favorable alternative among simple hypotheses satisfying the null hypothesis. The asymptotic distribution of these test statistics, under homogeneity, is found to be equal to the distribution of the likelihood ratio statistic. for testing analogous hypotheses about a set of means of normal populations. The analogous normal mean problem for test (iii) is considered. (i.e. test ~l ~ Pz ~ ... ~ PK against ~l < Pz ~ Us ~ ... ~ PK .)