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. Because the limiting distribution depends on the specific values of the unknown distributions under the null hypothesis, asymptotic and bootstrap approaches are proposed to overcome practical difficulties 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 finite sample sizes. The tests are applied to an example involving data for survival time for carcinoma of the oropharynx.