稀少性事件(rare events)是指發生可能性極低的事件。許多文獻中指出，在研究稀少性事件的資料中，當研究者探討二元反應變數(response)在稀少性事件之發生機率，會受何種因素影響時，最常使用的統計模型是機率單位模型(probit model, Bliss (1935))及邏輯式迴歸模型(logistic regression model, Berkson (
1944))。

　　在邏輯式迴歸中，King 和Zeng (2001)發現稀少事件(rare events)會降低迴歸中參數估計的準確性。King 和Zeng (2001) 進而提出兩種修正方法。近年來，多數學者選擇使用Firth (1993)所提出的懲罰最大概似估計量(penalized maximum likelihood estimator)。在事件之發生機率極低的情況下，Leitgöb (2013)將King 和Zeng (2001) 的修正方法及Firth (1993)的修正方式，進行電腦模擬比較，其模擬結果建議使用Firth (1993)的修正方式。
Elgmati et al. (2015) 進一步提出Firth (1993) 的懲罰最大概似估計方法的缺點。因此，他們提出了廣義的Firth 懲罰最大概似估計量(generalized Firth penalized maximum likelihood estimator)。

　　在以往的文獻中，因為許多學者主要針對邏輯式迴歸模型，在稀少性事件下，探討參數估計的影響及修正方法。由於在現有的文獻及程式軟體中，並沒有機率單位模型，在稀少性事件下，對參數估計的修正方法。因此，本論文主要針對機率單位模型，在稀少性事件下，探討對參數估計的影響及修正方法。模擬結果顯示，在不同$lambda$及樣本數下，使用廣義的Firth懲罰最大概似估計法後，估計量的偏誤會有大幅的改善。

This thesis studies the accuracy of parameter estimation of probit model with rare events. The ideas of Firth penalized maximum likelihood function and generalized Firth penalized maximum likelihood function for the logistics regression model are employed for probit model with rare events. The simulation results show that the bias of maximum likelihood estimator of regression parameters of probit model can be reduced by the penalized term in penalized likelihood function. Overall, when sample size is small, for example n=100, penalized MLE with = 0:1 performs better than other penalized MLE for event rate( π) < 0.01. While penalized MLE with = 0:5 performs better than other penalized MLE for event rate( π) > 0.05.

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