Statistical Inference For Nonparametric Regression Measurement Error Models

This paper is about statistical inference for nonparametric regression problems inwhich predictors are measured with error. In many areas of application, statisticallymeaningful models of interest are defined in terms of predictors that for some reason arenot directly observable, for example, sampling error, experimental error, instrumentalerror, error in survey, etc. In such situations, it is not uncommon for surrogate variablesto be observed instead. The surrogate variables for true predictors complicates the fun-damental statistical analysis on which the purpose of the analysis is inference about amodel defined in terms of true predictors. These problems are commonly called mea-surement error problems, and the statistical models and methods used to analyze suchdata are known as measurement error models or errors-in-variables models.This paper can be divided broadly into four main parts. The first part defines basicterminology of error model, data sources and the distinction between nondiferential anddiferential errors, and gives an overview of the basic ideas and estimation techniques ofregression model with measurement error.Nonparametric methods are enjoying increased application. The field of nonpara-metric regression has become extremely important in the past twenty years, and in mea-surement error problems techniques are now established. The second part briefly givesan overview of some important techniques in nonparametric regression models when thedata are observed with classical errors or with Berkson errors.The third part discusses the estimation of nonparametric regression models with theexplanatory variable being measured with Berkson errors or with a mixture of Berksonand classical errors. By constructing a compact operator, the regression function is thesolution of an ill-posed inverse problem, and we propose an estimation procedure basedon Tikhonov regularization.The fourth part of this paper considers estimation approaches for nonparametricregression measurement error models when both independent validation data on covari- ables and primary data on the response variable and surrogate covariables are available.An estimator which integrates orthogonal series estimation and truncated series approx-imation method is derived without any error model structure assumption between thetrue covariables and the surrogate variables. Most importantly, our proposed methodol-ogy can be readily extended to the case that only some of covariates are measured witherrors with the assistance of validation data.