This paper considers partial linear regression models when neither the response variable nor the covariates can be directly observed, but are instead measured with both multiplicative and additive distortion measurement errors. We propose conditional variance estimation methods to calibrate the unobserved variables. A profile least-squares estimator associated with the asymptotic results and confidence intervals is then proposed. To do hypothesis testing of the parameters, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties of the estimator and the test statistic are also established. 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. Estimation, hypothesis testing, and variable selection are discussed under the scenario of multiplicative distortion alone. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its applicability.
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