Statistical Inference For Nonparametric Time Series In A Locally Stationary Context

Time series analysis is an important subject in statistics.The establishment and improvement of the statistical theory of time series not only have important theoretical significance,but also have a very profound impact on practical applications.The traditional time series theory is based on the assumption that the observation series is stationary.However,in practical applications,the assumption of stationarity is often failed for many time series,such as many high-frequency data in the financial field(including the trading volume,the stock prices,etc.)are non-stationary.As a special kind of non-stationary time series,locally stationary process is not stationary from a macroscopic point of view,but from a microscopic point of view,it can be approximated by a specific stationary time series,thus it has good statistical properties and is more in line with many time series in real world,and has received extensive attention from many researchers in recent years.The analysis methods of time series include parametric methods and nonparametric methods.Although the theoretical properties of parametric methods are easier to prove,they are often based on some strong assumptions.Therefore,once the assumptions are not satisfied,the corresponding estimators are not efficient,or even incorrect.Nonparametric methods appeared in the application of time series in the 1940s and achieved good results,but they were not widely accpeted in academia at that time due to the lack of corresponding statistical theories.In recent years,as more and more researchers have begun to study the theoretical properties of nonparametric estimation in time series,nonparametric methods have not only established a solid theoretical foundation,but also obtained more opportunities to realize their potential in practice.Time-varying nonparametric models are an important type of structured nonparametric regression method,Vogt[81]used time-varying nonparametric regression models to fit locally stationary time series,and established the asymptotic properties for the corresponding kernel estimators.However,the time-varying nonparametric regression model cannot handle the case in which there are multiple explanatory variables influencing on the response variable.Therefore,combined with the idea of additive models,Vogt[81]also proposed time-varying additive models to deal with the multiple explanatory variables case better.In addition to the continuous-valued response variable cases,the discretevalued response variable cases are also very common in time series.The traditional linear regression and nonlinear regression methods can't be used to analysis the discrete-valued time series.As an extension of linear regression,the generalized linear model can be used to obtain the estimators of the unknown parameters by using maximum likelihood estimation or quasi-maximum likelihood estimation.It is one of the most important methods to analysis of discrete-valued response variables in statistics.However,the specific function form of the generalized linear model makes it impossible to get the efficient estimators like that of linear regression when the true regression function is different from the assumption.The generalized additive model,as a combination of generalized linear model and additive model,is essentially a type of semi-parametric model.Almost no assumptions are imposed on the form of the additive functions,except some continuous and differential conditions.Therefore,the generalized additive model can handle the complicated situations more flexibly.Through some nonparametric methods such as spline estimation and kernel estimation,not only the unknown function of the generalized additive model can be estimated,but also the asymptotic properties can be derived.Specifically,the dissertation is mainly divided into two parts.The first part takes the time series of continuous-valued response variables as the research object.Based on previous statistical theories,we further study the parameter estimation and asymptotic theory of time-varying nonparametric models and time-varying additive models under locally stationary background.We discuss how to use the structure of the locally stationary error series to obtain the estimators of the nonparametric functions,and demonstrate the consistency and the asymptotic normality of those estimators.The second part considers the time series of discrete-valued response variables.In the presence of periodicity,the generalized additive model is used to investigate the estimation of periodic terms and nonparametric functions.In the first chapter,we mainly introduce the background and significance of the of research,present the problems we studied,and review the literature in the related fields,and introduce the framework of the dissertation.In the second chapter,we introduces the basic concepts and tools used in the remaining chapters of the dissertation,including the definition of locally stationary process,local polynomial smoothing,B-spline,and the concept of generalized additive model.In Chapter 3,we study the time-varying nonparametric regression model with locally stationary errors.Although Vogt[81]investigated time-varying nonparametric regression models,they did not consider the structure of the error series.Therefore,the estimators they obtained are consistent but not efficient.We use the idea of two-step estimation to improve the estimators.Firstly,the response and the initial estimator of the time-varying nonparametric function are used to obtained the estimated error,which is used to fit the structure of the error.Next,the fitted structure of the error is used when when conducting the refined estimator of the time-varying nonparametric regression function,and we prove that the refined estimator is more efficient.At the same time,we also innovatively use the ULASSO method to simultaneously complete the model identification and coefficient estimation of the error.In Chapter 4,we discuss the estimation methods of time-varying additive model with locally stationary errors.Vogt[81]proposed a smooth backfitted algorithm for the time-varying additive model,but this method needs calculation at each time point,so the computation cost is high.In our dissertation,B-spline is used to obtain the initial estimate of the time-varying additive function,which greatly reduces the computation cost.Then we make full use of the estimators of the error structure to get the efficient estimators of the time-varying additive function using local polynomial smoothing.In Chapter 5,we consider the discrete-valued response of the time series.Under the background of the locally stationary explanatory variables,we use Bspline and kernel smoothing to estimate the trend and the additive component functions based on the generalized additive model.At the same time,the penalized maximum likelihood estimation helps us to correctly specify the period,and we obtain the(?)-consistency estimators of the period term.In the last chapter,we summarize the results of this dissertation,and present some potential prospects for future work.