Empirical Likelihood Ratio Test For The Stochastic Precedence Relationship Between Populations

The stochastic precedence relation between populations is an important quantity to describe the relation between two populations.The stochastic priority relation represents that one population trend is larger than the other and has been applied in many fields,such as upper tolerance limit,Behrens Fisher problem,ordered experimental analysis,reliability analysis,stress-strength test,etc.The research on stochastic precedence focuses on the distribution estimation under the constraint of stochastic precedence,but the hypothesis test of stochastic precedence has not been studied.In this paper,we mainly study the hypothesis test of stochastic precedence constraint.Under the condition that one distribution is assumed as a parametric model and both distributions are assumed as nonparametric,we construct an empirical likelihood ratio test using empirical likelihood.Under the original assumption of P(X?Y)= 1/2,we determine the asymptotic distribution of the empirical likelihood ratio statistics.When a population is assumed as a parametric model,the empirical likelihood ratio statistics asymptotically obey a mixed chi square distribution.When there is no parameter model assumption for the two distributions,the empirical likelihood ratio statistics asymptotically obey a weighted mixed chi square distribution.Our numerical simulation and analysis of oropharyngeal squamous cell carcinoma data and melanoma data show that the empirical likelihood ratio test can not only control the type I error,but also has a better test effect.In addition,in order to test whether the two populations satisfy the stochastic precedence hypothesis testing problem,the Permutation test based on the empirical likelihood ratio can well control the type I error rate and has a good testing power.