Estimation For Several Functional Models

Functional data refer to data in the form of functions such as curves,surfaces or anything else varying over a continuum,where a sample element is considered to be a function.For analysis of functional regression,the common assumption is that the error is independent and identically distribution(i.i.d.).However,dataset with structures such as dependent observations,partially observed functional ob-servations,outliers,etc,are also a common phenomenon.The aim of this thesis is to systematically investigate estimation of some functional regression models which can capture structure of dataset.In particular,three functional models are considered:functional linear model,partial functional linear model,varying coefficient model with functional response.Four structures are incorporated:er-ror term is a mixing sequence,error term follows ARCH model,dataset include outliers and functional observation with missing value in some subset of domain? mixing as one of the popular measures of dependence of sequence is usually adopted.For partial functional linear model(PFLM),when the errors forms a ?mixing sequence,we consider the estimation of the parameter and slope function of the model by functional principal component analysis(FPCA)method.The technique of Bernstein's big and small block is adopted to get the asymptotic normality of estimator of parameters.The effect of the a mixing dependence structure on the covariance of estimator for parameters is obtained comparing with i.i.d.error.The convergence rate of slope function is derived,and it is shown that it attain the optimal convergence rate.The finite sample performance is illustrated by simulated examples and a real application,which showed attain the expectationAutoregressive conditional heteroscedasticity(ARCH)model showing the het-eroscedasticity of the dataset is very popular in econometrics.For PFLM with ARCH(p)errors,we use a two step estimation method to get estimators.Firstly,we get the estimators of parameters and slope function,and get asymptotic nor-mality of parameters and convergence rate for slope function.Then least absolute derivation method is adopted to get the estimator of ARCH(p)model,and nu-merical studies show the robustness of proposed method.In further,asymptotic normality of the proposed estimators for ARCH(p)models is also derivedConsidered that not only the dataset is not i.i.d.,but also the functional observations is not perfect data such as some observations include missing value in subset of the domain.For functional linear model(FLM),when the errors is AR sequence and functional observations is only observed on a subset of the domain,we impute the score of the missing part firstly using the observed information in the view of mean square prediction error,and then give the estimate of FLM by classical FPCA method under L2 risk.Convergence rate is obtained demonstrating that it attains the minimax convergence rateFor varying coefficient model with functional response,the aim is to get the robust estimator,M-estimator,of covariate effect,which is not sensitive to outlier-s and heavy-tailed distribution of error.Asymptotic properties of the penalized M-estimator is established,and it is show that the estimator achieves the mini-max optimal rate of convergence under general loss function.Efficient algorithm alternative direction method of multipliers(ADMM)is used to compute objec-tive function to overcome the infeasibility and complexity of computation.The consistency of the ADMM is also derived.Simulation studies and diffusion tensor imaging data is conducted to show its finite sample performance.