Statistical Inference For A Functional Linear Model

In recent years,with the rapid development of computational science and cloud technolo-gy,developed information technology and high-performance computer are able to collect,store,transmit with time or space continuous curve and surface data.We collectively consider such da-ta as functional data.The concept of functional data was first proposed by Canadian statistician Ramsay in 1982.At present,functional data is widely found in the fields of engineering,social sciences and natural sciences.Specific disciplines such as finance,economics,environmental and medicine science have a wide range of applications,which we can give a few examples like atmospheric temperature curve data analysis,human growth curve data analysis,electricity load curve data analysis,etc.Using traditional principal component analysis or partial least squares to overcome the difficulties of dimensionality reduction for functional data may lead to some loss of information and lack of explanation.Statistical scholars have put forward various func-tional models to model the functional data in order to clear or ease the obstacles.Numerous research results have emerged.In this thesis,the statistical hypothesis test of the coefficient function in the functional linear model is mainly focused on,where both the response variable and the covariate are func-tional data.First of all,the thesis in Chapter 2 introduces the model of the function data in L2 space and two ways to smoothing functional data,including smoothing based on known basis functions expansions and smoothing through roughness penalty.In addition,Chapter 3 gives the representation of functional coefficient ?(s,t)in the function linear model using B-splines and proposes the constructive form of the test statistic for the hypothesis testing.Furthermore,statistical properties of the functional coefficient estimators are given.In Chapter 4,we use the Bootstrap stochastic numerical simulation procedure to assess the size and power performance of our proposed test.Last but not least,in Chapter 5 we apply the proposed method to the anal-ysis of sharing bicycle functional data.