Conditional absolute mean calibration for partial linear multiplicative distortion measurement errors models

In this paper we consider partial linear regression models when all the variables are measured with multiplicative distortion measurement errors. To eliminate the effect caused by the distortion, we propose the conditional absolute mean calibration, which avoids to use the nonzero expectation conditions imposed on the variables. With these calibrated variables, a profile least squares estimator is obtained, associated with its normal approximation based and empirical likelihood based confidence intervals. For the hypothesis testing on parameters, a restricted estimator under the null hypothesis and a test statistic are proposed. A smoothly clipped absolute deviation penalty is employed to select the relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. Lastly, a score-type test statistic is then proposed for checking the validity of partial linear models. We derive asymptotic distribution of the proposed test statistic. The quadratic form of the scaled test statistic has an asymptotic chi-squared distribution under the null hypothesis and follows a noncentral chi-squared distribution under local alternatives, which converge to the null hypothesis at a parametric rate. Simulation studies demonstrate the performance of our proposed procedure and a real example is analyzed as illustrate its practical usage.