Some Studies On Functional Regression Models

With the advance of technology and improvement of data collection,more and more functional data that are in the form of curves can be observed and recorded.Functional data have become a popular and important type of data.Functional data have applications in many subject areas,such as biology,chemistry,medical sciences,meteorology and economics.Functional regression models are becoming one of the most important tools in functional data analysis.Functional regression analysis poses challenges both for theory and computation due to infinite dimensional structures of functional data.On the other hand,the infinite dimensional structures of the data are rich in information,which bring many opportunities.Thus,Functional regression analysis becomes a very promising research direction.We focus on estimation and prediction problems for some functional regression models.The main content is as follows.(1)We study functional partially linear single-index model with covariates being functional covariates.We propose a profile least squares approach combined with local constant smoothing for estimating the slope function and the link function in the model.We demonstrate that our method enables prediction of the link function and estimation of the slope function with polynomial convergence rates.The convergence rate of prediction of the whole model is also established.Monte Carlo simulation studies show an excellent finite-sample performance.A real data example about average yield of oats in Saskatchewan,Canada is used to illustrate our methodology.(2)We propose semi-functional partial linear model with covariates being mixed data of functional covariate and scalar covariates.We propose a quantile approach combined with local constant weighted smoothing.We propose a three-stage estimation procedure for estimating the regression coefficient and the nonparametric function in the model.Under some regular conditions,we establish the asymptotic normality of estimators of regression coefficient.We also derive the rates of convergence of nonparametric function.Finite-sample performance of our estimation is compared with least square approach via a Monte Carlo simulation study.The simulation results indicate that our method is much more robust than the least square method.A real data example about spectrometric data is used to illustrate that our model and approach are promising.(3)We consider varying coefficient partially functional linear quantile regression model with covariates being mixed data of functional covariate and scalar covariates.We propose a quantile approach combined with local polynomial technology.We propose a two-stage estimation procedure for estimating the regression coefficient and the nonparametric function in the model.Under regularity conditions,we establish the asymptotic normality of the proposed estimator.We show that the estimated slope function can attain the minimax convergence rate as in functional linear regression.A Monte Carlo simulation study and a real data application suggest that the proposed estimation is superior to other existing models and methods.(4)We consider functional partial linear regression model.We propose a locally sparse estimation approach based on smooth and functional shrinkage technique.We can obtain the estimation of the model and identify null subintervals of the slope function simultaneously.Under regularity conditions,we establish the asymptotic normality of the estimated regression coefficient.We show that our method can identify null subintervals of the slope function with probability tending to one.Monte Carlo simulation studies show an excellent finite-sample performance,and the real data example about spectrometric data is used to illustrate that our approach is useful and promising.The estimation methodologies and conclusions in our research enrich the studies of functional regression models,which are also helpful to analyze the real problems in many application fields,such as biology,medical sciences and economics.