人類壽命資料分析經常發生截斷(truncation)的情況，因為觀測時間經常僅限於某個特定的區間內。這篇論文中主要針對有母數模型進行推論，在觀察樣本受限於雙尾截斷的情況下(也就是左尾截斷以及右尾截斷)進行分析。Efron and Petrosian (1999) 提出特殊指數族(special exponential family, SEF)為模型，特殊指數族的分布可定義如下： 其中 以及 。
其可配適雙尾截斷資料，但在Efron and Petrosian論文中，並無做進一步的理論及模擬探討，所以此篇論文中我們將雙尾截斷資料利用特殊指數族的理論及模擬完成。我們使用此模型為基礎，進而利用牛頓-拉弗森演算法(Newton-Raphson algorithm)以及固定點迭代法(Fixed-point iteration)得到參數最大概似估計量(Maximum likelihood estimator, MLE)，並且比較此兩種方法在參數估計上的優劣；為了確保三個參數的特殊指數族收斂性質，我們提出了隨機牛頓-拉弗森演算法(Randomized Newton-Raphson algorithm)。而在理論部分，若變數受限於雙尾截斷而導致隨機變數互相獨立但來自不同分配時，我們依然可以得到最大概似估計量的大樣本性質,諸如一致性(consistency)、有效性(efficiency)、漸近常態性(Normality)。最後我們利用人口壽命資料作為例證。

Truncation often occurs in lifetime data analysis, where samples are collected under certain time constraints. This thesis considers parametric inference when random samples are subject to double-truncation, i.e., both left- and right-truncations. Efron and Petrosian (1999) proposed to fit the special exponential family (SEF)

where and ,
for doubly-truncated data, but did not study it’s computational and theoretical properties. This thesis fills this gap.
We develop computational algorithms for Newton-Raphson and fixed point iteration techniques to obtain maximum likelihood estimator (MLE) of the parameters, and then compare the performance of these two methods by simulations. To stabilize the convergence under the three-parameter SEF, we propose a randomized Newton-Raphson method. Also, we study the asymptotic properties of the MLE based on the theory of independent but not identically distributed (i.n.i.d) random variables that accommodate the heterogeneity of truncation intervals. Lifetime data from the Channing House study are used for illustration.

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