A Likelihood Ratio Test Against Stochastic Ordering in Several Populations

The likelihood ratio test is often used to test hypotheses involving a stochastic ordering. Distribution theory for the likelihood ratio test has been developed only for two stochastically ordered distributions. For testing equality of distributions against a stochastic ordering in several populations, this paper derives the null asymptotic distribution of the likelihood ratio test statistic, which is characterized by minimization problems and has no closed form. A Monte Carlo simulation is conducted to study the limiting distribution. Since the limiting distribution depends on the speciic values of the unknown distributions under the null hypothesis, asymptotic and bootstrap approaches are proposed to overcome practical diiculties and implement tests based on the likelihood principle. Asymptotic validities for these tests are established and simulations are carried out to check their performances for nite Wright for helpful discussion and comments and Professor Frederick Williams for correction of the language. Special thanks are due to two anonymous referees, an Associate Editor and the Editor for their stimulating comments and regarding the simulation of the limiting distribution in particular. 1 sample sizes. The tests are applied to an example involving data for survival time for carcinoma of the oropharynx.