Several Statistics Models For Set-valued And Interval-valued Data

The presence of set-valued or interval-valued data is quite common in prac-tice.For example,the lesion images in medical imaging examinations are usually plane figures,which can be viewed as closed sets in R2.For another example,K-line chart,as an interval which contains the opening price,the highest price,the closing price and the lowest price of one day,is an important tool for decision making in the stock and futures markets.On the other hand,system complexity leads to data are ranges instead of real values.For instance,experts usually give the interval-valued data such as 3%to 6%,3.5%to 5.7%when they estimate the economic growth rate of China in next year.Due to the presence of interval-valued variables is quite common in practice,it is necessary to establish the statistical models of set-valued and interval-valued data for data analysis,inference and pre-diction.This dissertation mainly study several statistical models of set-valued and interval-valued data,which including interval-valued and set-valued nonparamet-ric regression models,constrained interval-valued linear regression model,interval-valued time series models and set-valued multivariate time series.Specific research contents are summarized as followsFor interval-valued and set-valued nonparametric regressions,this part first-ly discusses a nonparametric regression with interval-valued explanatory and re-sponse variables.The interval-valued nonparametric regression is fitted with center and radius,the estimators and their asymptotic properties are given with basis functions.A simulation study and a real data application show that the proposed model can fit one category data well.Next,a nonparametric regression with set-valued explanatory variable and scalar response variable is discussed.Combining the dp metric and asymmetrical kernel can estimate the unknown functional,which is a projection from set-valued space to real-valued space,the asymptotic property of estimations are also deduced.Simulation studies and real data analysis show the efficiency of the proposed model and approachFor constrained interval-valued linear regression model,this part is to split an interval into the left-and right-endpoints,and fit two dimension linear regression model with a constrained condition on response variables.Then proposes the three step estimation method which focuses on the bias-corrected and heteroscedasticity-adjusted modeling and weighted linear least squares with estimated covariance matrix.Theoretical conclusions and numerical evaluations show that the proposed estimator has higher efficiency than previous estimatorsFor interval-valued time series models,this part firstly gives the definition of generalized interval-valued Gaussian random variables,and further discusses the generalized interval-valued Gaussian white noise processes under two differ-ent assumptions.Then proposes the interval-valued mixed autoregression moving average model which bases on the assumption of the distribution of center and radius of error.The estimation,identification,prediction and diagnostic checking methods are discussed.Simulation study and real data analysis show the effi-ciency of the proposed model and approach.On the other hand,the constrained interval-valued autoregression model is discussed.The three step estimation,iden-tification,prediction and diagnostic checking methods are discussed.Simulation study and real data analysis show the proposed estimator has higher efficiency than previous estimatorsFor set-valued multivariate time series,this part mainly shows some pre-liminary theory and basic concepts.Firstly,this part gives the definitions and properties of set-valued vectors and set-valued random vectors.Then defines the set-valued multivariate time series and discusses the stationary property,expec-tation vectors,cross covariance matrixes,cross correlation matrixes and the opti-mal linear forecast of stationary set-valued multivariate time series.At last,the interval-valued vector autoregressive model and its coefficient estimations are dis-cussed,simulation study and real data analysis show the efficiency of the proposed model and approach...