This paper considers partial linear single-index 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. This method avoids to use the nonzero expectation conditions imposed on the variables in the literature. Using the calibrated variables, a profile least squares estimator is obtained. For the hypothesis testing of parameter, 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. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.

中文翻译：

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部分线性单指标失真测量误差模型的估计和变量选择

当所有变量都用乘法失真测量误差进行测量时，本文考虑了偏线性单指标回归模型。为了消除失真造成的影响，我们提出了条件绝对平均校准。该方法避免使用文献中强加于变量的非零期望条件。使用校准变量，获得轮廓最小二乘估计量。对于参数的假设检验，提出了原假设下的受限估计量和检验统计量。采用平滑剪裁的绝对偏差惩罚来选择相关变量。由此产生的惩罚估计量被证明是渐近正态的，并具有预言机属性。