對於具共變量誤差之存活資料，假設給定測量共變量，真實共變量之條件分佈(共 變量誤差分佈)已知，我們將研究柯斯正比風險模型中參數之半母數最大概然估計。我 們在一般的假設下，已得到模型參數可確認性，並且建立了半母數最大概然估計的存在 性。我們接下來的目的在於建立半母數最大概然估計的漸近性質，包含一致性、漸近常 態性、計算漸近共變異矩陣的漸近理論以及概然函數比的推論。我們將進行模擬試驗來 說明此一方法的數值表現。當研究資料中具有可確認之共變量資料或重複測量之共變量 資料時，我們也想將共變量誤差分佈與回歸係數，基線風險共同視為未知參數，研究其 最大概然估計的性質。 We will study semiparametric maximum likelihood estimators in the Cox proportional hazards model with covariate error, assuming that the conditional distribution of the true covariate given the surrogate is known. We have proved the model identifiability under regular conditions and established the existence of the SPMLE. Our next goal is to establish the asymptotic properties of the SPMLE, including consistency, asymptotical normality, asymptotic theory for the calculation of asymptotic variance, and inference for likelihood ratio test. We would also like to conduct simulation studies to demonstrate the performance of method. When validation data or replicate data are available, we attempt to model the covariate error distribution parametrically or nonparametrically and use the maximum likelihood principle to estimate regression parameter, baseline hazard, and covariate error distribution simultaneously. Further investigation in this regard will also be taken.
