Linear Covariate Adjustment Model Parameter Estimation

Linear regression model is one of the most important models in statistics,which is extensively applied in many areas.But in practice,the actual predictors and response are not observable.Instead,one observes contaminated versions of these variables, where the distortion is multiplicative,with a factor that is a smooth unknown function of an observed covariate.The simultaneous dependence of response and predictors on the same covariate may lead to artificial correlation and regression relationships which do not exist between the actual hidden predictor and response variables.So attention must be paid to this condition.The theme of this paper is to explore such confounding in regression and to develop appropriate adjustment methods.We demonstrate how the regression coefficients can be estimated.The proposed estimators are constructed in the following manner.First we establish a multiple varying coefficient model which connects with the CAR model and estimate the varying coefficients functionsβ_γ(·)in(2.2).The coefficients｛γ_γ｝_γ~p=0 are targeted in the second step,with weighted averages of the estimatedβ_γ(·),making use of the relations betweenβ_γ(·)andγ_γgiven by(2.1) and the identifiability conditions.The proposed methods are using the method of LP modeling for the estimation ofβ_γ(·)in the first step.But in this paper we propose two new estimation procedures where in the first proposed method,the local least squares is explored in place of smoothing splines.The thesis is organized as follows:(1) Looks back some backgrounds of the covariate- adjusted regression model,presents the previous investigations and covers some preliminary knowledge.(2) Proposes a new estimation procedure where in the first step the estimation procedures are based on B splines least squares and proofs the consistency and convergence rates of this estimator.(3) Proposes a new estimation procedure where in the first step the estimation procedures are based on B splines M-estimate and proofs the consistency and convergence rates of this estimator.(4) Proposes the asymptotic normality of the parameter estimators,and presents consistent estimates of the asymptotic variance,which are used for the construction of asymptotic confidence intervals for the regression coefficients.