Multiplicative Regression Models With Distortion Measurement Errors

When we deal with the measurement error data,the naive procedure by simply ignoring measurement errors always leads to a biased and inconsistent estimator.As a result,we should solve such practical problems by choosing some proper measurement error models.There are two types of measurement error data.One is the additive measurement error model.Another one has a multiplicative fashion,which we call the distortion measurement error model.In this paper we consider the distortion measurement error model.This paper studies the estimation and hypothesis test of multiplicative linear regression model with distortion measurement error.Through theoretical and simulation research,the estimation effect of the estimators are discussed.This paper considers estimation for multiplicative linear regression models when neither the response variable nor the covariates can be directly observed,but are distorted by unknown functions of a commonly observable confounding variable.After taking logarithmic transformation on the response variable,we propose an estimation methods for the parameter.That is the least squares estimator.Another is the least product relative error estimator without logarithmic transformation.For the hypothesis testing of parametric components,restricted estimators under the null hypothesis and test statistics are proposed.The asymptotic properties for the estimators and test statistics are established.A bootstrap procedure is proposed to calculate critical values.Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.The main work we have done is as follows:Firstly,we propose the product linear regression model with distortion measurement error and variable calibration process.We use the direct plug-in method(Cui et al.2009;Delaigle et al.2016;Zhao and Xie 2018)to obtain calibrated covariates and calibrated response variable.Secondly,by using the calibrated variables,we propose the LPRE estimator without logarithmic transformation.We consider statistical inference for??₀ to test whether??₀ satisfies some linear combinations or not.To mimic the null distribution of the test statistic,a bootstrap procedure is proposed to define p-values.Thirdly,we propose the least square estimation with logarithmic transformation.We investigate the large sample properties for the proposed estimators,test statistics and restricted estimators.And we prove the asymptotic properties of least square estimator and LPRE estimator by theory.Lastly,we conduct Monte Carlo simulation and real data analysis to illustrate our proposed methods.