Empirical Likelihood Estimation For Semi-functional Partially Linear Regression Model

Technological progress in collecting and storing is now providing datasets with functional features(such as curve and images,etc.)which are called functional data.Functional data can be found in many applied fields such as medicine,environmetrics,climate among others.Mathematicians especially statisticians are interested in studying functional data analysis,and there are many theory and applied research around functional data analysis.Many models have been developed and have been widely used,such as semi-functional partial linear regression model,and the research is still ongoing.Semi-functional partial linear regression model is an important statistical model in functional data analysis.It explores the correlation between scalar response variables and functional explanatory variables,and integrates the information of parametric and non-parametric models.The model is of adaptability and predictive ability.By using this model to process functional data,the advantages of the two models can be combined and a better fitting effect can be obtained.In practical applications,data missing due to various reasons is very common,and it is of great practical significance to study the semi-functional partial linear regression model in the case of missing data.This paper applies the empirical likelihood method to solve responses missing at random.Empirical likelihood method is often used in nonparametric statistical inference,including the construction of parameter confidence region and so on.For example,compared with the normal estimation method,the empirical likelihood method does not need to estimate the variance.On the other hand,the empirical likelihood method does not impose constraints on the shape of the confidence region,but the shape of the confidence region is determined by the data itself.Statisticians have used the empirical likelihood method in linear regression model,generalized linear model and other models,but this method has not been more developed in semi-functional partial linear model.The main contents of this paper are how to apply empirical likelihood method to semi-functional linear model when the responses missing at random,and using k-nearest neighbor(KNN)method to select the bandwidth of kernel function in the model.The main research content is divided into two parts:First,empirical likelihood statistical inference of semi-functional partial linear model.The kernel function of the model is constructed by KNN method,and then the estimator is constructed by empirical likelihood method.The theoretical results and proof show the theoretical significance of empirical likelihood method.The simulation part verifies that the method has a certain practical value.Second,empirical likelihood statistical inference based on the response missing at random in a semi-functional partially linear model.When the responses missing at random,the estimators of parametric and nonparametric regression operators are constructed by empirical likelihood method,and the asymptotic properties are given.Then the simulation results are compared with the least square method,which shows the advantages of this method and gives the real data analysis.