Statistical inference for linear regression models with additive distortion measurement errors
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Statistical inference for linear regression models with additive distortion measurement errors Zhenghui Feng1 · Jun Zhang2
· Qian Chen3
Received: 11 August 2018 / Revised: 16 October 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract 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. Keywords Measurement errors · Kernel smoothing · Linear regression models · Residual based estimator · Model Checking Mathematics Subject Classification 62G05 · 62G08 · 62G20
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00362018-1057-2) contains supplementary material, which is available to authorized users. Extended author information available on the last page of the article
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1 Introduction Many empirical data sets are contaminated by the errors-in-variables. The problem of measurement errors is one of the most fundamental issues in empirical data analysis in numerous fields, such as medical research, health science, and economics. Measurement error is common in many disciplines owing to inappropraite instrument calibration, among many other factors. In cases in which some variables have been measured with errors, estimation by using the observed errors-in-variables may cause large bias and lead to inconsistent estimates even with large samples. For example, measurement errors in simple linear regression lead to underestimation of the coefficient, which is known as attenuation bias (Fuller 1987). In nonlinear models, the structure of the bias is more complicated (Carroll et al. 2006). The classical statistical estimation and inference for measurement error mode
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