Adjusted Empirical Likelihood Test Under Inequality Constraints

At present,many studies are based on the assumption that the parameter space involved is the entire space,which does not impose any restrictions on the relevant parameters of statistical inference.However,it is clearly difficult to meet the complex testing requirements On the contrary,it is worth noting that when conducting parameter testing on a certain type of population,there will always be some testing requirements,which often manifest as certain restrictive conditions,and inequality form constraints are more common.That is to say,parameter space related problems with constraints should also be taken seriously in practical applications,and under today’s mature statistical inference technology,the dimension and breadth of testing requirements will only be continuously amplified In contrast,constrained parameter spaces also have the characteristics of providing accurate parameter effective information and promoting the development of optimization problems in various fields,demonstrating extensive practical value.Studying them has important theoretical and practical significance.However,in the case of constrained parameter space,there are few studies on statistical inference using nonparametric methods.Although many scholars have conducted detailed research,such as Chen Li made empirical likelihood inference of many models under inequality constraints,and obtained excellent properties of relevant statistics,there is still room for improvement in the testing power of small samples.Compared with the empirical likelihood method,the adjusted empirical likelihood method proposed by Chen et al.has the advantages of consistent properties under large samples and slightly better under small samples.It also properly solves the convex hull problem of empirical likelihood,and is a practical non parametric statistics method Therefore,based on a comprehensive analysis of previous research results,this article uses the adjusted empirical likelihood method to study the population mean model,linear model,and mixed effects model with inequality constraints in the parameter space in three modules.The research work of this article can be summarized in the following two points:1.Using the adjusted empirical likelihood method,the population mean model was first studied,with unilateral and bilateral tests conducted.The bilateral tests included bilateral tests with inequality constraints and conventional bilateral tests,both of which constructed the adjusted logarithmic empirical likelihood ratio test statistic and proved that its limit distribution was a mixed weighted chi square distribution;Secondly,univariate linear models and multivariate linear models were subjected to unilateral hypothesis testing and conventional bilateral testing for regression coefficients with inequality constraints,respectively;Finally,a unilateral test with inequality constraints was conducted for a simple mixed effects model.2.Some parameters in the linear model and mixed effects model have been estimated using least squares,and it has been proven that the limit distribution conclusions of the statistics under these two models are consistent.Their limit distributions are both mixed weighted chi square distributions At the same time,under different population distributions or error distributions,numerical simulations are presented from multiple perspectives of each statistic to compare the results with other existing research methods,verifying the feasibility of the proposed method.