Interval Prediction For Nonparametric Regression Models With Fixed Design Based On Spline Method

The estimation methods of nonparametric regression models have plentiful theoretical developments,which provide an excellent foundation for the application of the models.Prediction is an important aspect in the applications of nonparametric regression models.However,the prediction methods research of nonparametric regression models is rather backward.To further explore such issues,I discussed the properties and interval predication of spline estimation about nonparametric regression models under fixed design.The details as follows:Firstly,the nonparametric regression model with ?-mixing errors under fixed design points is considered.The regression function and variance function in the model are estimated by the spline method.The consistency and convergence rate of the estimators are discussed.The estimated mean and standard deviation of unknown functions' Square-Root of Average Squared Error(RASE)decrease with the increase of sample size in numerical simulations.Meanwhile,empirical analysis also proved the feasibility of this estimation method.Secondly,the interval prediction of the parametric models is extended to the nonparametric models,and the applications of three prediction methods based on spline estimation,namely non-extrapolation method,linear extrapolation method and nonlinear extrapolation method in the interval prediction of the nonparametric regression model with homoscedasticity are considered.By analyzing the mean interval coverage probability(MICP)and average interval width(AIW)predicted by the three methods in different situations,the advantages and disadvantages of the three prediction methods are compared.The numerical simulation results show that the MICP of the linear extrapolation prediction method is closest to the given confidence level and the AIW is the smallest under moderate sample.In addition,the nonlinear extrapolation method is the second.The empirical analysis results show tha t the MICP values of the extrapolation prediction method are closer to the given confidence level.Therefore,the linear extrapolation method is an appropriate interval prediction method.In the case of the application of the three prediction methods in nonparametric regression model with heteroscedasticity,the results of numerical simulation and empirical data analysis show that under moderate sample size,the linear extrapolation method is closer to the confidence level.As far as the AIW is concerned,the linear extrapolation method is superior to the non-extrapolation method on a whole.For different samples,the linear extrapolation method is more stable and better than the non-extrapolation method and non-linear extrapolation method.