除此之外，我們也利用Empirical Process Theory及Approximately Least Favorable Submodel的相關理論來證明迴歸係數(Regression Coefficient)和致病傾向參數(Frailty Parameters)之NPMLE的漸近常態性(Asymptotic Normality)及漸近有效性(Asymptotic Efficiency)。為了求取Approximately Least Favorable Submodel，我們計算迴歸係數和致病傾向參數的Efficient Score Function。此Efficient Score Function為一積分方程的解。這裡，我們利用此積分方程中相關函數(為一Banach Space 上的Operator)的解析性，以Functional Analysis為主要工具證明Efficient Score Function的存在性。
最後，對於迴歸係數和致病傾向參數的假說檢定，我們證明在虛無假說下，Profile Likelihood Ratio Statistic的漸近分布是具有自由度3的卡方分布。對此，我們便可求得迴歸係數和致病傾向參數的信賴區域。

The identifiability of the parameters and the existence of NPMLE are established under certainregularity conditions. In addition to the asymptotic consistency, the asymptotic normality and efficiency of the NPMLE for the regression coefficient and frailty parameters are proved, and a convergence rate of the NPMLE for the baseline cumulative hazard function is established.
The profile likelihood ratio statistic for hypothesis testing and the related confidence regions for the regression coefficient and frailty parameters are also studied.

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