Based On The Functional Model Of Data Detection And Estimation Theory

With the rapid development of science and technology during the past twenty years, people encounter a lot of the data with functional feature (referred to as functional data). It is very necessary to develop functional data analysis because functional data analysis has many unique advantages. For example, it can carry on statistical analysis for the data which come from the infinite dimensional functional space. More information can be dig by functional data analysis. It allows which the observation object has different observation times. The method still apply for non-functional data, and so on. At the same time, functional data analysis has a wide application in growth analysis, meteorology, biomechanics, economics etc. Functional data analysis has become a hot field of statistics during the recent twenty years. Many researchers have been devoted to this aspect and has achieved many results in theory and application. However, due to the infinite dimensional feature of functional data, it has also encountered great challenges for statistical inference of functional data. Therefore, research is still in its infancy on the functional data, many problems still need further research.In this paper, we mainly consider the function regression model. In recent years, many researchers proposed some regression models with functional data. But most of them focus on the functional linear model and function nonpara-metric model. Therefore, this paper tries to enrich and develop some functional regression model and puts forward the idea of model detection.First, we use the method of functional principal component and Group Lasso to detected the significant order of function polynomial model. We apply the proposed method to the spectral data and found some very interesting conclusion. Then, we develop a functional polynomial regression models with auto-regression error due to the actually needing. We are facing two Problems for the proposed model, namely:detect significant order of functional polynomial regression model and detect significant order of auto-regression error. We develop a joint detection method, it can simultaneously solve the two problem.Secondly, we develop the function polynomial multiplication model. It is a useful alternative for the model of Yao and Muller (2010). We extend the least absolute relative errors criterion (LARE) to the estimation of the model and de-velop the new method to detect the order of functional polynomial multiplicative model. We also apply the proposed the model and methods to the Canadian climate data and some good results are obtained.Thirdly, we consider the partially linear model for functional data. The number of observations for each individual is sparse in the longitudinal data analysis. However, it is possible that some subjects are densely observed while others are sparsely observed in practice. Moreover, in dealing with real data, it may even be difficult to classify which scenario we are faced with and hence to decide which methodology to use. So we establish a unified estimation method for partially linear model for sparse functional data and dense function data and some large sample properties are obtain.Finally, we establish functional additive models based on the functional sin-gular component analysis when both covariates and the response are functional data. Principal component analysis can only reflect the own information of the covariates and response variables and ignore their dependencies because it base on the decomposition of their auto-covariance function. However, cross-covariance function of the covariates and response variables is decomposed for function sin- gular value analysis, it better reflect their dependencies. The estimators and asymptotic properties are investigated for the build functional additive models.