我們介紹一個半母數擴充風險模型 (semiparametric extended hazards model)，它包含比例風險模型 (proportional hazards model) 和加速風險模型 (accelerated failure time model)。根據擴充風險模型這項特性，我們可以建立針對比例風險模型或加速風險模型的適合度檢定 (goodness-of-fit test)。另一方面，當比例風險模型和加速風險模型均不適合所描述的資料時，擴充風險模型提供了在模型選擇上的另外一種選擇。對具有與時間不相關風險因子的存活資料，我們藉由找出一核心平滑的簡化概似函數 (kernel-smoothed profile likelihood function) 的極大值對此擴充風險模型提出一個新的估計方法。透過一概似函數比值檢定 (likelihood ratio test)評估比例風險函數和加速風險函數建立一個適合度檢定。最後結果的估計量會被證明在大樣本時具有強烈一致性且服從常態分配，此外，也指明出概似比檢定統計量的分配。接著，我們考慮擴充風險模型的一推廣，其中允許風險因子的觀測值可以隨時間而改變。我們使用計數過程法 (counting processes approach) 重新建構擴充風險模型，並提出一不偏估計函數族 (a class of unbiased estimating function)。所得的估計量在大樣本以及適當條件下時具有一致性且服從常態分配。

We introduce a semiparametric extended hazards (EH) model which includes the proportional hazards (PH) model and the accelerated failure time (AFT) model as special cases. By the nested structure of the EH model, one can construct a goodness-of-fit test for the PH model or the AFT model. On the other hand, the EH model provides another choice of model for the given dataset when the PH model and the AFT model both fail to fit. For the survival data with time-independent covariates, we propose an estimation for the EH model by maximizing a kernel-smoothed profile likelihood function, and evaluate the goodness-of-fit of the PH model or the AFT model through a likelihood ratio test. The resulting estimators are proven to be strongly consistent and asymptotically normal, and the asymptotic distribution of the likelihood ratio test statistic is identified. Moreover, we consider an extension of the EH model in which the covariates are allowed to be time-dependent. We use the counting processes approach to reformulate the EH model and propose a class of unbiased estimating functions for the estimation in the EH model. The resulting estimators are proven to be consistent and asymptotically normal under regularity conditions.

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