Statistical Inference Of Semi-functional Linear Model With Autoregressive Errors

Some data are recorded continuously according to time or space points in clinical medicine,finance,and other fields.They can be regarded as a function of time or space and called functional data(FD).In the traditional functional data analysis,it is usually assumed that the random errors are independent and identically distributed(i.i.d.),but this assumption may be unreasonable in some cases(for example,the annual electricity consumption data).Therefore,this dissertation studies the parameter estimation,error autocorrelation test,and robust estimation of a semi-functional linear model under the assumption that the random error obeys autoregression(AR(p)).The model is not only the generalization and development of the independent identically distributed random error,but also can enhance the interpretability of the model.Therefore,its research is meaningful both in theoretical research and practical application.Specifically,the primary research of this dissertation can be summarized into the following three aspects:(1)Assuming that the random errors are serially correlated,a semi-functional linear model with autoregressive error is proposed,and the estimation of model parameters and nonparametric functions is studied.First,based on Mercer’s theorem,the slope function is reduced by using the functional principal component basis expansion.Second,the nonparametric function is approximated by B-spline basis expansion.Finally,the estimation of the model is obtained by a two-step iterative algorithm.We also establish the large sample properties of estimators of the nonparametric function,slope function,and autoregressive coefficient under some regular conditions.Simulation research and real data analysis verify the effectiveness and rationality of the proposed model and estimation method.(2)Test whether there is sequence correlation in random error.Based on the empirical likelihood method,an empirical likelihood test statistic is proposed,which can test whether the random error is independent and identify the order of autocorrelation if the serial correlation holds.The proposed empirical likelihood statistic does not need to estimate variance because it is data-adaptive and has a nonparametric Wilks’ theorem.Under the null hypothesis and certain regularity conditions,it is proved that the test statistics asymptotically obey the chi-square distribution.Simulation research and example analysis verify the effectiveness of the proposed test statistics.(3)If the random error distributions follow a heavy-tailed distribution,the robust estimation method of the semi-functional linear model with autoregressive error is studied.The estimation of traditional linear regression models is mostly based on the least square or likelihood method.However,the least square estimation is very sensitive to outliers.Especially in the case of the heavy tail of error sequence,the efficiency of parameter estimation may be reduced,and the corresponding estimation will no longer be effective.Therefore,aiming at this problem,this dissertation introduces robust loss functions(such as Huber and absolute loss function)to obtain the robust estimation method of the model.Simulation research and example analysis show that the robust estimation has good statistical quality as OLS if the random error follows the normal distribution.However,if the random error is a heavy tail,the robust estimation performs better than the OLS estimation.