Estimation Of The Measurement Error Regression Models With Index Variates

Since the 1980s,the semiparametric regression model has become the mainstream in the statistical regression model research.It not only combines the characteristics of parameteric and nonparametric regression models,but also reduces the data dimen-sion which can overcome the "dimension disaster".Therefore,it has more adaptability and robustness,and has extensive application prospect.The important semiparamet-ric regression models include:partially linear model,single-index model and varying coefficient model,partially linear single-index model and partially varying coefficient single-index model,etc.Among them,the studies of single-index models by projecting the high-dimensional covariates to a one dimensional linear space,which can avoid the"dimension disaster" in the multivariate nonparametric regression effectively.In practice,the data always cannot be measured accurately.For example,there exists measurement errors due to various reasons,such as:the precision of the measur-ing tool is not enough or the operation is not standard,etc.In the statistical analysis,the models with measurement errors are called the measurement error models.Cur-rently,most of the studies are considering the covariates with linear measurement error,i.e.,the observable substitution variables W and the true covariates X satisfy the relation:W = X + U,where U are measurement errors with mean zero.In this paper,we mainly study the partially single-index measurement error mod-el.It contains the two cases of nonlinear part of the covariates and all of the covariates are measured with errors.Firstly,we estimate the parameters and nonparametric functions by combining the simulation-extrapolation(SIMEX)method with the oth-er classical estimation methods,and give the asymptotic properties of the estimates under appropriate conditions.Then,we validate the large sample properties of the estimators by some simulation experiments.Further,this paper also studies the more general partially varying coefficient single-index model with measurement errors in the index covariates.Finally,we verify the effectiveness of the proposed estimation of the three measurement error models by practical examples,respectively.