Statistic Inference Of Semi-parametric Partially Linear Varying Coefficient EV Model With Surrogate Data And Validation Sample

The semi-parametric model is very important in statistical study.It contains two parts:nonparametric part and parameter.And partial linear models both have the strongpoint of linear regression and the advantages of nonparametric regression.Therefore,this model has attracted the high attention of many experts,and it is wide-ly applied in many areas.The wide application of linear model is due to its good model interpretability.By contrast,non-parametric model is more robust;The semi-parametric model is more flexible than the parametric model and it contains many kinds of models,such as partially linear varying coefficient model,single index mod-el,multiple index model,partial linear single index model.Therefore,semi-parametric regression can effectively avoid the "curse of dimensionality" problem than multiple parametric regression.In practical,because of the limitations of experimental conditions,we can only get the observations instead of the true values.There exists errors between the true value and observations and the errors are inevitable.So,this”measurement error"problems have aroused much attention of many scholar.We call this kind of error-in-variable model EV model.In this article,we study the relationship of surrogate data and validation samples.In the practical application,obtaining long term variable information is not realistic and We can only choose variables measured easily in this way in the later period.There must exist errors,but the errors structure may not be a simple additive model structure.It may be very complex and one solution is to use validation method.In this paper,we mainly study the semi-parametric varying coefficient error-in-variable model with surrogate data and validation sample,where both co-variable in parametric part and response variable have measured errors.We do not assume any error structure.Firstly,based on validation data sets,we use the kernel regression technique to incorporate the information contained in both surrogate data and real data.Then the parameter? and non-parameter ?(.)are estimated by using the semi-parametric profile least square method.what's more,We conduct GLR test for varying coefficient function.Secondly,three important bandwidth selection methods are given.Then,the estimator of ? and a(·)under the hypothetical conditions are proved to be asymptotically normal.Finally,simulation studies are conducted to illustrate the finite sample properties of the proposed estimators and efficiency of GLR method.