Abstract
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.
Highlights
In many applications involving regression analysis, observations of the variables of interest may include with measurement errors
For the distortion functions (ψM (·), ψA(·)), the multiplicative distortion function ψM (·) is a p × p-diagonal matrix given by diag ψM,1(·), . . . , ψM,p(·), and the additive distortion function ψA(·) is a p-dimensional vector given by ψA,1(·), . . . , ψA,p(·) T
Regarding additive distortion, [33] proposed a residual-based estimator of the correlation coefficient between two unobserved primary variables, and showed that the estimator is asymptotically efficient as if all the variables are observed exactly, i.e., without distortion. [34] studied the estimation and variable selection in partial linear single-index models when the response variable and some covariates are measured with additive distortion measurement errors, i.e., φM (u) ≡ 1 and ψM,r(u) ≡ 1, r = 1, . . . , p
Summary
In many applications involving regression analysis, observations of the variables of interest may include with measurement errors. [34] studied the estimation and variable selection in partial linear single-index models when the response variable and some covariates are measured with additive distortion measurement errors, i.e., φM (u) ≡ 1 and ψM,r(u) ≡ 1, r = 1, . This paper intends discusses partial linear models that contain both multiplicative and additive distortion measurement errors. In [1], the authors only considered the existence of multiplicative distortion measurement errors (φA(u) = ψA,r(u) ≡ 0), and used the conditional mean calibration procedure to estimate (φM (u), Y, ψM,r(u), Xr): Y. All technical proofs of the asymptotic results are given in the appendix
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