Statistical Inference For Spatial Autoregressive Model With Functional Covariates And Its Application

With development of economic and scientific technology,huge amounts of data can be easily collected and stored.In particular,some types of data are observed in high dimensions and frequencies containing rich information.We usually call them functional data.As we known,functional linear model and its derivatives are widely used models for functional data sets.However,the models usually make assumptions that the samples are usually independent with each other.However,in some applications,the independent assumption is usually violated especially for spatial data because there may exit some connections between countries such as geographical connection or trade relationship.In order to better model such nearby correlated information,we adopt the style of autoregressive model,namely,adding nearby response variables as explained variable.Consequently,based on above discussion,we focus on the statistical inference for the following functional spatial autoregressive model and its derivatives.The first chapter proposes the spatial autoregressive quantile model with functional covariate.We use functional component analysis combined with instrumental variable to derive the consistent estimators for slope function and its spatial correlation parameter.Under regular conditions,we present the optimal convergence rate of the estimator for slope function and its asymptotic normality for spatial correlation parameter.In simulation study,we compare our estimation methods with other methods such as generalized method of moments,ordinary quantile regression approach.The results show the priority of our estimation method.Lastly,we apply the proposed model to Growth data set and obtain the explain in economic aspect,which validates the appropriate model.The second chapter studies spatial autoregressive model with functional covariate,called functional linear spatial autoregressive model.Compare with the model in chapter two,we add the other nonfunctional explain variables.Based on functional component analysis and generalized method of moments,we produce the consistent parameter estimators and the convergence rate of slope function estimator.Based on consistent estimators,we construct a test statistic of the residual sums of squares under null and alternative hypothesis.Under regular conditions,we establish the asymptotic properties of the parameter estimators and proposed test.In addition,we extend the model to functional partial linear spatial autoregressive nonparametric model.The simulation results are consistent with the theoretical part.At last,we use proposed model for the Growth data and get its residuals,which indicate that the proposed model is appropriate.The third chapter focuses on spatial autoregressive single index model with functional covariate.We use B-spline method combined with profile generalized method of moments to obtain the asymptotic normality for parameter estimators and the optimal convergence rate for nonparametric and slope function estimators.Based on the sparsity of parameter variable,we propose its sparsity estimators and its asymptotic properties.The theorems are in consistent with simulation results.At last,we use the model to stock return of a company to illustrate the suitable model.Functional spatial quantile autoregressive model extends the functional linear model and spatial autoregressive model and gives the parameter estimators given different quantile levels for spatial autoregressive model with functional covariate.Functional linear spatial autoregressive model assumes that the linear correlation between functional covariate and other nonfunctional covariates.Under regular conditions,we give its statistical inference.Functional spatial autoregressive single index model uses B spline approximation with profile generalized method of moments to produce the consistent estimators for single index parameter estimators and its sparsity estimators.The proposed estimation methods and its results enrich the studies of functional spatial autoregressive model,which can be used to deal with the practical problems in economics,meteorology,biology and so on.