Research On Asymptotic Properties Of Error Density Estimator And Error Distribution Estimator In Autoregressive Models

Autoregressive time series model,as one of the basic tools of time series analysis,has been concerned and studied by many economists and statisticians.One not only pay attention to the parameter estimation of autoregressive model,but also pay attention to the error estimation of the model,and the latter is very important for the study of model diagnosis and prediction.Therefore,this thesis,based on the dependent error,studies the consistency of error density estimation and error distribution estimation of autoregressive models.The residual kernel density estimators and residual kernel distribution estimators are constructed,and the limit distribution and uniform convergence rate of these estimators are obtained.Moreover,simulation studies and a real data analysis are conducted to investigate the performance of the residual kernel density estimators.This thesis mainly completes the following research work.In Chapter 2,we consider the error density estimation in the first-order autoregressive models with positively associated or negatively associated random error.Under mild regularity conditions,some asymptotic normality results of residual density estimator are obtained when the autoregressive models are stationary process and explosive process.In addition,in order to illustrate these results,some simulations such as fitting curve,confidence intervals and integrated mean square error are provided.It shows that the residual kernel density estimator is better than the residual histogram density estimator.In Chapter 3,we consider the error distribution estimation in the firstorder autoregressive modes with positively associated or negatively associated random error.Under some regularity assumptions,some asymptotic normality results for residual distribution estimator are obtain when the autoregressive models are stationary process and explosive process.Similarly,some simulations of estimated curves and integrated mean squares error are performed.It shows that the residual kernel distribution estimator is better than the residual empirical distribution estimator,and the residual kernel distribution estimator is smooth.As an application,the prediction about the Gross Domestic Product(GDP)by residual kernel distribution estimator in autoregressive models is given.In Chapter 4,we study the problem of error density estimation in order nonlinear autoregressive model with -mixing error terms.Combined with the error dependence and the stationary properties of the nonlinear autoregressive model of order ,the asymptotic normality and uniform convergence rate of error kernel density estimation are given.Further,the finite sample properties of residual density estimator are investigated in various models through Monte Carlo simulations.The results also show that the residual kernel density estimator is more suitable for estimating the error density of the model than the residual histogram density estimator.As an application,the self-threshold autoregressive model is used to study the annual average number of sunspots.Finally,we summarize the research work about the error density estimation and error distribution estimation of the first-order autoregressive model and the error density estimation of the -order nonlinear autoregressive model,and list some plans for future research.