Empirical Likelihood Inference Method For Semi-parametric Varying-coefficient Partial Linear Model

In many fields of human society,such as biomedicine,financial economy,industrial production and so on,regression model is often used to reveal the quantitative relation between variables.Therefore,a number of important regression models have been proposed and developed.Because each field has its own characteristics,so in practice,there are often missing data,measurement error data and other complex data we may encounter.Therefore,it is of theoretical and practical significance to study the statistical inference of model under complex data.Empirical likelihood method is a nonparametric statistical inference method proposed by Owen.Compared with the traditional method,empirical likelihood method has many advantages.For example,it does not need to construct the pivot statistics,nor to constrain the shape of the confidence domain.The shape of the confidence domain is completely determined by the data itself.In this paper,we mainly study the empirical likelihood inference of partially linear single-index varying-coefficient model under the circumstance of complete data,data with missing at random and data with measurement error.It mainly contains the following contents:In the second chapter,the empirical likelihood method is applied to the statistical inference of partially linear single-index varying-coefficient model.By introducing an auxiliary random vector and using the empirical likelihood method,we propose two construction methods of confidence region of two interest parameters.Under certain conditions,we can prove the asymptotic distribution of the empirical log-likelihood ratio statistics.The first empirical likelihood ratio constructed by us asymptotically converges to the weighted sum of several independent chi square variables with degrees of freedom of 1.Because of the need to estimate the weights,the accuracy of the confidence region will be reduced.Then we put forward an adjusted empirical likelihood ratio,and prove that the adjusted empirical log-likelihood ratio is asymptotically obey the standard chi square distribution,and then through this property we give the confidence region of interest parameters.The results of numerical simulation further verify that the statistical inference method based on empirical likelihood method proposed in the second chapter has fine finite sample properties,which proves the theoretical results and practical value of the method in confidence region construction.In the third chapter,the empirical likelihood method is extended to the statistical inference of partially linear single-index varying-coefficient model with missing covariates at random.We combine the inverse probability weighting method with the empirical likelihood method,and put forward an adjusted empirical likelihood inference method.It is proved that the adjusted empirical log-likelihood ratio is asymptotically subject to the standard chi square distribution,and the confidence region of interest parameters is given by this property.In the numerical simulation,we compare the adjusted empirical likelihood method with the unadjusted empirical likelihood method,and find that the adjusted empirical likelihood method is better in the case of limited sample set.In the fourth chapter,the empirical likelihood inference of single-index varying-coefficient partially linear EV model is also investigated.We introduce a corrected empirical likelihood ratio function and derive the Wilk's phenomenon under certain conditions.In this chapter,we also verify the applicability and goodness of our method by numerical simulation.