Estimation For Single-index Varying-coefficient Models With Multiplicative Distortion Measurement Errors

In the era of big data,how to perform statistical analysis on massive data with multiple data types is a challenge we currently face.In the process of statistical inference of these data,the limitations of parametric regression models and non-parametric regression models have become more obvious.To better describe the relationship between variables,statisticians proposed semiparametric regression models which combine the advantages of these two types of models.The single-index varying-coefficient model is an important model in the semi-parametric regression models,and it can degenerate into a single-index model or a varying-coefficient model under some conditions.Therefore,it is necessary for us to study the statistical inference of the single-index varying-coefficient model.In the medical and economic fields,variables are often affected by measurement errors during observation.It is difficult to directly observe the true values of these variables containing measurement errors.However,classic semi-parametric regression models usually assume that the variables can be accurately observed.At this time,if we ignore the influence of measurement errors and use the classic semi-parametric regression models to make statistical inferences,then the obtained results will be biased or even completely wrong.For more accurate inference results,we need to calibrate the variables containing measurement errors.In this paper,we consider the estimation for single-index varying-coefficient model with both the response variable and the covariate containing multiplicative distortion measurement errors.Firstly,we use the conditional absolute mean calibration method to obtain nonparametric estimators of the unknown distortion functions.Next,divide the observed value of the variable with the obtained estimator of the distortion function to get the calibrated variable,that is,an estimator of the true value of the variable.Then,we use the single-index varying-coefficient model to analyze the calibrated variables,and obtain the estimators of the model by the estimating equation method.Finally,we provide the asymptotic properties of the estimators and their proof results.Besides,we construct two numerical simulation examples to verify the superiority of the proposed corrected estimating equation method and analyze the body fat dataset.