K-nearest Neighbours Estimation Of Functional Nonparametric And Semi-parametric Model With Responses Missing At Random

The development of modern science and technology provides the technical support for huge amounts of data collection and storage,and functional data analysis(FDA)method,which is based on the functional characteristics of high-dimensional data,has been widely used in environmental science,chemistry,biology,economics,hydrology and other fields,and is a hot research direction of statistical circles in recent decades.Functional nonparametric regression model and semi-functional partially linear regression model are common in the research of functional data statistical inference in recent years,and among them,semi-functional partially linear regression model as a special kind of semiparametric model,mainly combine nonparametric regression model with the parametric parts which is easier to explain,thus the adaptability of the model as well as the ability to predict is increasing.The k-nearest neighbor(kNN)method is also a powerful tool in functional data analysis.The local adaptive bandwidth is selected by the discrete integer parameter k replacing the continuous parameter h in kernel regression estimation,which will greatly improve the calculation speed of the model.However,due to various factors such as objective machines factors and subjective human factors,the missing data widely exists in all walks of life,and how to deal with the missing data has a great impact on the application of statistical methods in the real field.The data missing at random has been widely studied in finite dimensional situation,but it haven't got more development in functional situation.How to combine kNN estimation with the functional nonparametric /semiparametric model in the case of the response variables missing at random is a huge challenge.In this thesis,the main work is constructed the corresponding kNN estimators and proved the asymptotic properties of the estimators theoretically based on the functional nonparametric/semi-parametric model.Simulation experiments and real data analysis further verify the good prediction effect of the estimators.The main research content is divided into the following two parts:(1)kNN estimator of functional nonparametric regression with responses missing at random and its applicationIn the case of responses missing at random,the estimator of nonparametric regression operator with functional explanatory variable is obtained through kNN regression estimation,and the asymptotic property of the estimator is proved theoretically.Secondly,the simulated curves of different sample sizes were used to verify the influence of sample size and the missing rate on the prediction results,and the advantages of kNN estimation compared with kernel regression estimation are demonstrated.Finally,the feasibility and predictive ability of the model are tested through the prediction results of the temperature curves for PM2.5 content in Beijing,and the practical significance of constructing the estimator with the kNN method is clarified.(2)kNN estimation in semi-functional partially linear regression model with responses missing at randomSemi-functional partially linear regression model is actually an extension of the nonparametric model,the mainly work of the fourth chapter is to further construct the kNN estimators in the case of responses missing at random,and after sorting out the proof procedure in detail of the asymptotic properties of the estimator of the unknown parameter factor and the uniform almost convergence rate of the unknown regression operator,the simulation experiments is further verified the influence of sample size and the missing rate on the prediction results.Finally,compared the predict results in semi-functional partially linear regression model with which is in nonparametric model is demonstrating the good prediction effect.