Empirical Likelihood Inference For Nonlinear Regression Models

The nonlinear regression model can be used to explain accurately the re-lationship between data. It can be used widely in many scientific research,especially, in the field of physics, chemistry, biology, engineering and so on.Therefore, it is necessary to study the estimation of nonlinear regression model,which is an important part of dealing with regerssion problems and plays animportant role in theory and practice. Although plentiful results on the non-linear regression model have been obtained in current literatures, most resultsmainly focus on the estimators of parametric components and their asymp-totic properties. In order to increase the accuracy of estimators of parameters,the confidence regions of parameters often need to be constructed which is themajor works of this dissertation.Empirical likelihood proposed by Owen(1988) is a nonparametric methodof statistical inference, it has many advantages on the construction of con-fidence regions. Besides the range preserving of regions, transformation in-variance, Bartlett correction and no need for constructing a pivotal statistic,in particular, empirical likelihood dose not involve variance estimation andthe shape and orientation of confidence regions based empirical likelihood aredetermined entirely by the data. Since the empirical likelihood method wasproposed, it has aroused great interests of statisticans. Empirical likelihood has been successfully applied to a large class of statistic models and manyfields.In chapter 2, the nonlinear errors-in-variables regression model is dis-cussed in detail. The empirical log-likelihood ratio statistic for the unknownparameters is proposed. In order to overcome the computationed difficulty,we also propose a smulation-based empirical log-likelihood ratio statistic forthe unknown parameters. It is shown that the two proposed statistics havethe asymptotic chi-squared distributions under some suitable conditions, andhence they can be used to construct the confidence regions of the parameters.At the end of this chapter, we also discussed the efficiency of the confidenceregions of the unknown parameters which is obtained by the smulation-basedmethod.Chapter 3 deals with the nonlinear semiparametric regression model, andconstructs the empirical log-likelihood ratio statistic for the unknown parame-ters. It is shown that the proposed statistics have the asymptotic chi-squareddistribution, and hence it can be used to construct the confidence regions ofthe parameters. In addition, the least squares estimator of unknown parameteris constructed and its asymptotic behavior is proved. Finally, the accuraciesof confidence regions are compared under different error variance by means ofthe simulation study. We also compare the proposed method with the least squares method in terms of the coverage accuracies and average lengths of theconfidence intervals. Simulation results indicate that our method performsmuch better than the least squares method.