Statistic Inference Of Single-Index Varying-Coefficient Model With Measurement Errors

Regression model has been playing an important role in statistics because of its wide application.As one kind of the regression models,the semi-parametric regression model has the characteristics of both parametric regression model and non-parametric regression model,which fully excavate the hidden information in the data and become the mainstream model in the field of statistics.The single-index varying-coefficient model is one of the semi-parametric regres-sion models,it combines the characteristics of the single-index model and the varying coefficient model,avoids the common problem of”curse of dimensionality" in the mul-tidimensional parameter,and makes the single-index model popularized.It is also easy to be explained in practical applications.However,in the process of solving prac-tical problems,we will find that variable data can not be accurately measured and the true value can not be obtained under the influence of various reasons.The causes of this phenomenon are:sampling error,non-standard measuring tools,non-standard operation and soon.In statistical research,we call the data with measurement error as "measurement error data" and the problem with measurement error as "error-in-variables model".Until now,the problem of additive measurement error is one of the most studied problems in statistics,and most of the existing literatures mainly study the problem of covariates with additive measurement errors.In this paper,a single-index varying coefficient model with measurement error is studied,which includes the situation where the index covariates with measurement error and all variables with measurement error.First,we obtain the estimation of the coefficient function and the index parameter by using the(SIMEX)method,the local regression method and the estimating equation method.Then,under appropriate conditions,we prove the large sample properties of the estimators.Finally,through numerical simulation and example analysis,we verify the effectiveness of the estimation method.