An Estimation For Functional Linear Model With Dependent Errors

With the development of technology,functional data widely exist in engineering technology,social science,natural science and other fields.It is widely used in finance,economy,environmental science,medicine and other specific disciplines.In functional data analysis,functional linear model is the most important and concise model of functional data modeling.The estimation method of traditional functional linear model focus on mean regression.Besides,traditional functional linear model always assumes that the random errors are independent and distributed(I.I.D).However,in practical problems,the random error may be dependent,and it is not appropriate to put the I.I.D assumption.As is well known,quantile regression has been widely used as a robust alternative to mean regression analysis.However,the efficiency of the quantile estimation is susceptible to the specific value of the quantile.The composite quantile estimation method combines multiple quantile information,which is more effective with single quantile information.In this paper,we study the estimation of functional parameter and error parameter for functional linear models under dependent errors.In addition,we propose composite quantile regression for functional linear model with dependent data,in which the errors are from a short-range dependent and strictly stationary linear process.In the second chapter,we introduce the basic knowledge of functional data,the method of functional principal component analysis and composite quantile estimation method.Then we start from the functional principal component analysis,and use the sample covariance operator and the mean function to obtain the parameter estimator of the functional linear model with GARCH errors in the third chapter.The convergence rate of parameter estimator is then calculated.Moreover,we estimate the parameter of the GARCH model based on the regression residuals with the least absolute deviation method.The obtained asymptotic distribution of the parameter estimator can be found in this chapter.After that,the performance of the proposed estimation methods for the finite sample is verified by random simulation.And then in chapter4,the functional principal component analysis is employed to approximate the slope function and functional predictive variable respectively to get an estimator on the slope function,and the convergence rate of the estimator are obtained under some regularity conditions.Simulation studies are presented for illustration of the performance of the proposed estimator.Finally,in chapter 5,some practical examples are given to evaluate the effectiveness of the proposed method.