Empirical Likelihood Inference For Semiparametric Regression Models With Nonlinear Function

Semiparametric regression model with nonlinear function has the ability of the interpretability of parametric model and the flexibility of nonparametric model,in addition to, it can overcome the limitations of linear functions in traditional semiparametric model. In practice, missing data, measurement error data and other complex data are often encountered. Empirical likelihood method, as an important nonparametric method, has been widely used for constructing confidence regions of interesting parameters. The result of research on empirical likelihood inference for semiparametric regression model with nonlinear function under some complex data has certain theory value and broad application prospect. This thesis mainly studies empirical likelihood method for partially nonlinear model and varying coefficient partially nonlinear model, and testing for nonstationarity of spatial-temporal varying coefficient model. The contents of the thesis are as follows:(1)Empirical likelihood inference for partially nonlinear models is studied firstly. For unknown parameter and nonparametric function, empirical loglikelihood ratios is proposed, respectively. Asymptotically chi-square distribution is proved and then the corresponding confidence region for unknown parameter and simultaneous confidence bands for nonparametric function can be constructed.The maximum empirical likelihood estimators of the parameter and nonparametric function are proposed and the asymptotic normality are obtained. A simulation study and a real data indicate that, compared with normal approximation-based method, the empirical likelihood method performs better in terms of confidence intervals of parameter and confidence bands for nonparametric function.(2) The estimation of confidence regions for partially nonlinear models under some complex data is studied. Partially nonlinear models when the response variables are missing at random is concerned. To avoid the estimation of unknown weights or an adjustment factor with classical method, a bias-corrected imputation technique to improve the accuracies of confidence regions is proposed. Partially nonlinear models when the explanatory variables and response variables are measured with error are considered,respectively. Under the help of surrogate variable and validation data, two estimators of unknown parameter in nonlinear function are obtained. Empirical log-likelihood ratio for unknown parameter is constructed and has asymptotically distributed as weighted sums of independence chi-square.Simulation study indicates that, the empirical likelihood method performs better than the normal approximation-based method in terms of confidence intervals.(3)Empirical likelihood inference for varying coefficient partially nonlinear model is studied. An empirical log-likelihood ratio for the unknown parameter in nonlinear function is defined. The proposed statistic is shown to be asymptotically chi-square. The maximum empirical likelihood estimators for the parameter is obtained and its asymptotic normality is proved. Varying coefficient partially nonlinear model with additive measurement errors in the nonparametric part is studied. The local corrected profile nonlinear least square estimation procedure for parameter in nonlinear function is proposed and then the asymptotic normality properties of the resulting estimators is established. In addition to, the local bias corrected empirical log-likelihood ratio statistic for the unknown parameter is defined and it is shown that the statistic is asymptotically chi-square distribution under some suitable conditions. Simulation studies show the effective of the proposed method.(4) Testing for nonstationarity of spatial-temporal varying coefficient model is studied. Based on the estimated residuals, likelihood ratio statistics are constructed for testing the temporal and spatial nonstationarity of the regression relationship and the p values of the test statistics are calculated with the bootstrap method. Then, appropriate statistics for testing the temporal and spatial nonstationarity of the estimated coefficients are proposed and the p values are calculated with the third-order moment χ2approximation method. Simulation studies and real example show that the test methods are valid.