Statistics And Inference Of Dynamic Single-index Variable-coefficient Models

Single index coefficient model is an important semi-parametric model,which is widely used in the fields of economics,finance,and biomedicine.The parameter structure of single index model is a linear combination of partial variables,while the non-parametric structure can flexibly describe the nonlinear relationship between variables.In practice,some scholars have found that the parameters in the single index coefficient model may change with time,space or other variables.To overcome this shortcoming,an effective method is to allow the coefficient of the single index coefficient model changes with the variable of interest,thus we can directly depict the dynamic dependence between the response variable and the predictors.We call the model that meets this feature as the dynamic single index varying coefficient model.Intuitively,the dynamic single index varying coefficient model combines the advantages of varying coefficient models and single index models.It can not only dynamically describe the relationship between the response variables and the predictors,but also include as many relevant variables as possible to reduce modeling bias.At present,related studies on the dynamic single index varying coefficient model is rare.Generalized varying coefficient model with unknown link function is a special case of dynamic single index varying coefficient model.Hitherto,the related literature focuses on the analysis of independent and identically distributed data or functional data,mainly discussing the construction of estimators and the asymptotic properties.However,there are some shortcomings in the theory analysis of dynamic single index varying coefficient model,such as the convergence rate of the estimator based on polynomial splines is lower than the optimal non-parametric convergence rate,or only the point by point nature is obtained,which makes people cannot understood the dynamics structure of the varying coefficient.On the other hand,there is a lack of research on dynamic single index varying coefficient models in model testing,variable selection,etc.The main contribution of this paper is that we study of dynamic single index varying coefficient model under independent and identically distributed data and local stationary processes systematically for the first time.The statistical inference theory includes establishing the estimation method and the asymptotic properties of estimators,testing the applicability of the model,making model selection to improve the accuracy and effectiveness of the model,and we also study the practical application of the proposed model.The first chapter mainly introduces the research background and significance,and reviews related research on dynamic single index varying coefficient models,local stationary processes,methods of non-parametric estimation and variable selection.At the same time,this chapter explains the research content,article structure,and innovation of this article briefly.The second chapter introduces the estimation method and model selection method of the dynamic single index varying coefficient model under independent and identically distributed data.Since the dynamic single index varying coefficient model cannot be identified,we propose a method for model identification based on spline approximation,and establish spline estimates of varying coefficient component functions and index component functions.Under appropriate conditions,we prove the consistency and asymptotic normality for the estimators.It is worth noting that the convergence rate of each component function estimate can reach the optimal nonparametric convergence rate.In order to describe the estimation of each component function more accurately,we establish a simultaneous confidence band.In addition,we construct the L2 distance statistic to test the applicability of the proposed model,that is,whether the dynamic single index varying coefficient model can degenerate into a single index coefficient model.We also establish the asymptotic normality of the test statistic.In order to solve the dimensional disaster problem of non-parametric models,we use the SCAD penalty to select important variables and prove the consistency of the penalized estimators.In numerical simulations,we study the finite sample performance of the estimator and the power of the test statistics.Finally,we apply the proposed model and methods to the analysis of body fat data and Boston house price data.The third chapter extends the dynamic single index varying coefficient model to quantile regression.As we all know,when the data contains outliers or the error is a thick-tailed distribution,directly applying the conditional mean regression model in the second chapter will result in estimation bias.However,the objective function is not differentiable under quantile regression,which makes it difficult for us to make statistical inference.In Chapter 3,we proposed a three-step estimation method based on splines smoothing.Under appropriate conditions,we proved that each component function estimates are consistent,and for the first time,the asymptotic normality of each component function estimate was established under quantile regression.In addition,we use the SCAD penalty to identify whether the interaction effects between variables are dynamic,that is,to test whether the variable coefficient component function is constant,and to prove the consistency of the penalized estimates.We have studied the finite sample properties of the estimators through numerical simulations and NO2 data analysis.In Chapter 4,we study on the dynamic varying index coefficient model under local stationary time series.At present,many scholars have studied the nonparametric modeling of time series and its methods,but most literatures are based on the stationarity assumption.However,in economics,finance and other fields,the stationary assumptions may be too harsh and untenable.For example,variables such as per capita income and consumption index are varying with time.This chapter weakens the stationary assumptions to locally stationary process,which has been widely discussed as a non-stationary process.Chapter 4 extends the models of the previous two chapters by allowing the varying coefficient function to change with the index.It is a broader class of dynamic single index varying coefficient models.In this chapter,we mainly study the varying index coefficient models under local stationary time series,and analyse the statistical inference theory of the model.We use a spline backfitted kernel algorithm to obtain the estimates of the index component function and the varying coefficient component function.Under appropriate conditions,we proved the consistency and established an asymptotic normal distribution for the estimators.However,the estimates of the varying coefficient component has an additional bias term due to the influence of the index function.In addition,we constructed a simultaneous confidence band of the varying coefficient component function.We use the Bootstrap method to test whether the varying coefficient component is truly time varying.Finally,numerical simulations confirm that the proposed method has good performance under finite samples,and we use the model in this chapter to analyze Hongkong environmental pollution data.In Chapter 5,we make a summary and reflection of this paper,and point out some shortcomings.