Some Studies On Functional Regression Models With Interactions

With the advance of modern technology,more and more data are being recorded continuously during a time interval or intermittently at discrete time points.Both of them are functional data.Functional data has become an important area in statistical research.As a type of sparse functional data,longitudinal data has experienced a long-run development due to its widespread application in medicine and other fields.Incorporating interaction improves interpretability.Sometimes the interaction itself is what people are interested in,for instance covariate-specific treatment e?ect is actually the interaction between covariates and indicator of exposure to treatment.It is necessary to fill the gap in functional regression models with interaction,although it will increase model complexity.The content of this thesis is as follows.(1)We study the prediction of generalized functional linear model with single-index interactions.We reduce the dimensionality of the functional predictor using functional principal component analysis(FPCA).We fit the model using an iterative procedure by minimizing a local quasi-likelihood using truncated FPCA series.By treating the number of FPCA scores as a tuning parameter and allowing it to diverge to infinity,we show that,for a wide range of this truncation number and di?erent bandwidths used by the nonparametric component in the single-index interaction,the parametric component of the model is root-n consistent and asymptotically normal.In addition,the overall prediction error is dominated by the estimation of the nonparametric function in the single-index interaction: an outcome that leads to a CV-based procedure to select the tuning parameters.We also show that the prediction error in the functional e?ect enjoys the minimax optimal rate in Cai and Hall(2006).In a crop yield prediction application,we show that our single-index interaction model yields lower prediction error than the conventional functional linear model and other competing nonlinear functional regression models.(2)We extend the generalized functional linear model with single-index interactions to the case of multiple functional predictors.We use multivariate FPCA to reduce dimensionality of the multivariate functional predictor.We fit the truncated model using a MAVE-based generalized quasi-likelihood function.Assuming the truncation parameter diverges to infinity,we establish the asymptotic normality of parametric and nonparametric estimates for a wide range of truncation parameter and di?erent bandwidths.The theoretical result of overall prediction error holds in multivariate functional predictor case.The findings in simulation and real data analysis are consistent with the theoretical results,which implies that the CV-based procedure for tuning parameter selection still apply.The model is finally applied to crop yield data set,and further decreased the prediction error.(3)Treatment selection based on patient's personal information has been widely recognized in modern medicine.In this paper,we consider a generalized partially linear single-index mixed e?ect model for longitudinal clinical trial or observational data,in which treatment e?ect is modeled by a semiparametric single-index term.This model allows treatment e?ect varying with patient's multiple baseline or timevarying characteristics.An approach combining local linear regression,penalized quasilikelihood and local penalized quasilikelihood is proposed to derive parametric and nonparametric estimates.An appropriate algorithm is also presented for practical implementation.We establish asymptotic normality for the estimators.A simulation study is conducted to evaluate finite sample performance of the proposed method.The methods and findings in this thesis enrich the research of functional regression models with interactions,and are helpful to real data analysis.