Heterogeneous Structure Inference Based On Empirical Likelihood Method In Linear Model

In statistics and economics,the problem of change point has been a major concern of experts since it was first introduced.The reason is that this problem is relevant to our life.There are numerous articles on the estimation of the location of change point,but no one has studied the situation of single change point in the random error sequence of the piecewise linear model by the empirical likelihood method.In this paper,the empirical likelihood method is used to study the problem.The paper investigates the problem of detecting the change point of error distribution in linear model and the problem of estimating single change point in error sequence.In order to clarify whether there is piecewise difference between the errors of the piecewise linear models,this paper establishes a hypothesis testing method based on the idea of empirical likelihood.In this part,we first give the piecewise linear model and the hypothesis testing problem.Meanwhile the residuals is obtained by least square method,and the asymptotic property of the empirical distribution of residuals is studied.Secondly,When the null hypothesis holds,we construct the empirical likelihood ratio function by using the empirical distribution of errors.And the asymptotic distribution of the empirical likelihood ratio function is obtained.In fact,errors are not observable,we can not get the empirical distribution of errors,but we can get the residuals,and then get the empirical distribution of the residuals.Therefore,the measure we take is to replace the errors with the residuals.Similarly,when the null hypothesis is established,we use the empirical distribution of the residuals to establish the empirical likelihood ratio test statistics.Finally,Monte Carlo simulation is carried out to prove the correctness of the conclusion.The results show that the asymptotic distribution of the statistics based on residuals constructions performs well under the null hypothesis when the sample size is large.For the case that there is a single change point in the error sequence of linear model,we first use the empirical likelihood method to find the estimator of the real change point.Secondly,we study the large sample properties of the change point estimator.When the sample size tends to infinity,we prove the consistency of the change point estimator(?)and(?)respectively,and calculate the convergence rate.Finally,we get the accuracy of the estimator of change point by Monte Carlo simulation,which proves that our estimation method is feasible.