| Functional data analysis is widely used in biostatistics,chemometrics,econometrics,medical research and other fields.Since functional data are infinite dimensional by nature,classical statistical analysis methods under multivariate observation are no longer applicable,so the statistical analysis of functional data has attracted much attention in recent years.We find that most of the statistical analyses of functional models in the literature use functional principal component analysis,and only consider finite and fully observed data.In this paper,we study the problem of estimation of functional regression models under incomplete data,where we use B spline method and reproducing kernel Hilbert space method to study the estimation of unknown functions,observations are missing at random,and it also involves the problem of high-dimensional estimation.The specific study includes the following aspects:1.We investigate the empirical likelihood of the parameters in a single-index functional partially linear model with the absence of response variables or/and partial covariates.Based on the approximation of the linkage and slope functions by B spline basis functions,the empirical likelihood ratio function of the model parameters is constructed,and its asymptotic distribution is proved,and the maximum empirical likelihood estimator of the parameters is further defined and its asymptotic normality is established.The selection of variables for the model parameters is also discussed,and the Oracle property of the parameter penalty estimator is proved.The feasibility of the proposed estimation method is investigated using Monte Carlo simulations.2.For a functional partial linear regression model with linear process errors,under missing observations,the estimates of the parameters and slope functions in the model are constructed by the Hilbert space regeneration kernel method and the inverse probability weighting method,and their convergence rates and asymptotic normality are proved.Based on the SCAD penalty method,the penalized estimates of the parameters are defined and their Oracle properties are established.At the same time,the test statistics of the linear hypothesis of the parameters are constructed based on the penalty estimates of the parameters,and their asymptotic distribution is proved.The finite sample performance of the proposed estimation method is studied by simulation and pork data were analyzed by using the proposed method.3.For the single-index functional partial linear regression model with highdimensional data,the B-spline method is used to estimate the link function,and the reproducing kernel Hilbert space method is used to deal with the slope function.In the case of randomly missing data,the unknown functions in the model and the estimators of the parameters are constructed,and their convergence rate and asymptotic normality are investigated.Further,the SCAD penalty method is applied to construct the penalty estimates of the model parameters.The variable selection of the model is studied and its large-sample property is demonstrated.The simulation study shows the feasibility and validity of the proposed estimation method.The main innovations of this paper are:(1)we improve the classical minimization estimation problem for single-index functional partially linear regression models with complete data to constructing the empirical likelihood ratio function of the model parameters and defining its maximum empirical likelihood estimates with missing data at random.(2)we extend the study of functional partial linear regression models under the assumption of independence to model errors to a non-independent linear process,and improve the principal component analysis method commonly used in the literatures into a reproducing kernel Hilbert space method on the estimator for the slope function.(3)we extend the single-index functional partial linear regression model under finite-dimensional and complete data to the high-dimensional case and improving the estimation of the partial slope function for functional data from the principal component analysis method to reproducing kernel Hilbert space method. |
