Empirical Likelihood Inference Of Functional Single-Index And Partially Linear Single-Index Model

With the advantage of the era of big data,there is a lot of data in all walks of life that we need to analyze and process.Faced with a large amount of data and a wide variety of data,multivariate data analysis has been widely used.In multivariate data analysis,sample data generally has discrete and finite features.But the information collected by modern data collection techniques includes not only the data processed by traditional statistical methods,but also the data generated by processes with functional forms.For example,an automatic data collection system,etc.In addition,in some research areas,the sample data obtained is a curve or other functional image.Data with this characteristic is called functional data,functional data refers to data that varies with a contiguous set(time,space,etc.)in a variety of forms,but the process of generating data is composed of functions.This type of data analysis is suitable for a variety of problems that are difficult to solve in scalar and vector frameworks in biology,medicine,meteorology,Econometrics,finance,chemistry,and physics and many other subject areas have a wide range of applications.A large number of references have been used to study the functional linear regression model in detail.The main research object is the functional single index model and the partial linear single index model.For functional single-index models,the commonly used analysis methods are the contour least squares method and the two-stage method.This paper uses empirical likelihood method,and empirical likelihood is one of the important methods of statistical inference.The empirical likelihood method has many advantages over the existing statistical methods.The shape of the empirical likelihood region is only related to the sample.The empirical likelihood method is superior to the least squares method in calculating the confidence interval length of the parameters and the size of the coverage probability.The main content of the paper is:In the part one.Firstly,a brief overview of the research background and significance of this paper.Secondly,Introduce a systematic single-index model and empirical likelihood based on existing domestic and foreign related literature.Finally,outline the main content,key points and innovations of the paper.The part two is statistical inference based on function-based single-index model and function-based partial linear single-index model.Firstly,the Karhunen-Loeve(KL)expansion is used to discretize the single index part of the model,and then the auxiliary random vector is introduced to further construct the estimation equation of the single index coefficient function to perform empirical likelihood inference,and the logarithm is empirically obtained.Since the estimation equation contains two unknown parameters: the link function and its first derivative,the local linear smoothing method is used to estimate the two parameters,and the secondary kernel and the corresponding optimal bandwidth are selected.The large sample properties of the estimated parameters and the confidence domains of the constructed parameters are obtained under given assumptions,and theoretical proofs are given.Finally,simulation analysis and empirical analysis.In the last part,we summarize the research results of this paper and propose what can be further studied in future work.