Effcient Inference For Complex Longitudinal/Functional Data

With the development of the society and modern science and technology,lon-gitudinal data and functional data analysis have been widely used in biomedicine,psychology,economics and environmental science.Longitudinal data and function-al data are typical repeated-observation data,such data are measured at discrete time points with certain observation errors.Therefore,it is realistic for this paper to analyze longitudinal data and functional data in a unified framework.We assume that there are n subjects being observed,and we denote the number of measurements within subjects as n_i,i=1,...,n.Depending on whether each n_iapproaches infinity,dense data and sparse data are studied separately as two antithetical data types.The methodologies used to treat the two situations have been different in the literature.Based on some existing studies,this paper will study the nonparametric modeling and statistical inference of longitudinal/functional data under the unified framework of sparse data and dense data.The first chapter is the introduction,which mainly introduces the research back-ground of longitudinal/functional data,as well as the structure and innovation of this paper.In Chapter 2,we study the time-varying additive model(tv AM)of longitudi-nal/functional data,which is effective at avoiding effectively the curse of dimension-ality and capturing dynamic features.We propose a unified spline-backfitted kernel estimators of the tv AM with sparse or dense longitudinal or functional data.Those estimators are both computationally expedient so they are usable for allowing very high-dimensional covariates,and theoretically reliable so inference can be made on the component functions with confidence.Under the setting of sparse data and dense data,we prove the two-step estimators have the same asymptotic distribution as that of oracle estimators.When it is uncertain whether the data is sparse or dense,we also establish a unified asymptotic normality,and thus construct a unified asymptotic confidence interval of the unknown function.Also,a testing statistic that can adapt to the sparse and dense cases in a unified framework is proposed to check whether the bivariate nonparametric functions are time-varying,and the asymptotic distribution of the proposed test statistic is derived.Simulation studies are conducted to assess the finite-sample performance of the proposed model and methods,and two different types of data are considered to illustrate the proposed method.Quantile regression models the effects of covariates through conditional quantiles of the response variable,rather than the conditional mean,which makes it possible to characterize any arbitrary point of a distribution and thus provide a complete descrip-tion of the entire response distribution.Compared to the classical mean regression,quantile regression is more robust to outliers and the error patterns do not need to be specified.Similar to Chapter 2,we study the statistical inference of longitudi-nal/functional data under the unified framework of sparse data and dense data with quantile regression model in Chapter 3 and Chapter 4.Specifically,in Chapter 3,we study the varying-coe cient quantile regression model and propose a local polynomi-al method suitable for both sparse and dense longitudinal/functional data.To reduce the computational burden caused by the non-smoothing estimating equations,we u-tilize a kernel function to approximate the quantile score function,which results in smoothing estimating equations.We prove that the asymptotic theories are different based on a sparse or dense assumption,depending on many unknown factors.Based on the smooth estimation equation,we directly construct the”sandwich”formula of the asymptotic variance and propose a unified asymptotic normality.In addition,we also obtain the one-step Newton-Raphson estimation of the coe cient function,which further simplifies the calculation.We prove that the one-step estimators are consistent with the fully iterative estimators.We exhibit the finite-sample perfor-mance of the proposed method and asymptotic theory through numerical simulation and two examples of real data.In Chapter 4,we discuss the time-varying additive model with quantile regres-sion.Similar to the second chapter,we first construct the spline estimators of each component function and trend function,and then use the kernel smoothing method to obtain the unified weighted local linear estimators of each unknown function.D-ifferent from the traditional quantile regression,we use convolution-type smoothed objective function in the second step.We prove the asymptotic normality of”oracle”estimators of trend function and component function,and the consistency between the two-step estimators and”oracle”estimators with sparse and dense data.In addition,based on the smoothed quantile regression,we directly use the standard sandwich formula for variance estimation.so as to construct unified asymptotic properties.Finally,we show the rationality of the model,estimation method and asymptotic theory proposed in this chapter through numerical simulation and examples of UK climate data and economic growth data.In the fifth chapter,we give some conclusions of this paper,and make further prospects for the future work.