Statistical inference for linear regression models with additive distortion measurement errors

We consider estimations and hypothesis test for linear regression measurement error models when the response variable and covariates are measured with additive distortion measurement errors, which are unknown functions of a commonly observable confounding variable. In the parameter estimation and testing part, we first propose a residual-based least squares estimator under unrestricted and restricted conditions. Then, to test a hypothesis on the parametric components, we propose a test statistic based on the normalized difference between residual sums of squares under the null and alternative hypotheses. We establish asymptotic properties for the estimators and test statistics. Further, we employ the smoothly clipped absolute deviation penalty to select relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. In the model checking part, we suggest two test statistics for checking the validity of linear regression models. One is a score-type test statistic and the other is a model- adaptive test statistic. The quadratic form of the scaled test statistic is asymptotically chi-squared distributed under the null hypothesis and follows a noncentral chi-squared distribution under local alternatives that converge to the null hypothesis. We also conduct simulation studies to demonstrate the performance of the proposed procedure and analyze a real example for illustration.