Statistical Inference For Semiparametric Partially Linear Models With Additive And Nonadditive Errors

The semiparametric regression model plays a very important role in the statistical theory. In order to make full use of the relevant information in the data, the model in-troduces not only the parametric component,but also the nonparametric function. This practice greatly optimizes the fitting effect of the model. As one of the semiparametric models, the partially linear model combines the characteristics of the linear model and the nonparametric model, not only has the characteristics of easy explanation of the linear model, but also has the robustness and flexibility of the nonparametric model. More importantly, the model can avoid the problem of’curse of dimensionality’. These features make the model more widely applied in practical problems.But in practical problems, we usually can not directly get the true value of the variables, the data obtained is the observed values of variables, these observations often exist in a variety of errors, such as the error caused by sampling, the tool used in the experiment or the error of the operation and so on. We refer to the observed data as proxy variables of real, the problem of covariates with measuring error is called ’measurement error problem’, the statistical models and methods which is used to fit measuring error data is called’errors in variables model’.In this paper, we mainly study the semiparametric partially linear model with additive and nonadditive measurement error, in which the observed value of the pa-rameter part contains the nonadditive measurement error, and its true value needs to be estimated with the auxiliary variables, the observed value of the nonparameter part contains the additive measurement error. We first calibrate the parametric covariates, and then use the semiparametric least-square method to estimate the parametric com-ponent and nonparametric function. Under appropriate assumptions, the asymptotic normality of the parametric component nonparametric function and the error variance is proved, we also propose the GLR test for the parametric components. Finally, nu-merical simulation and practical examples are given to verify the proposed estimation method and test method.