混合長尾損失分配的尾部純保費估計

保險損失分配選擇之適切性，與分配參數估計之準確性，皆為純保費訂價上的重要關鍵。保險損失分配為右偏之長尾分配，傳統上，我們利用對數常態分配、柏拉圖等分配來建構損失模型，但對數常態分配、柏拉圖等分配在估計尾部機率上常有高估或低估之情況發生，因而使得尾部純保費之訂價風險提高。McNeil (1977) 和 Resnick (1977) 利用廣義柏拉圖分配(GPD)來估計損失模型尾部之情況，其結果較傳統模型為佳。因此，本研究將在一混合指數(EXP)與 GPD 分配的混合參數模型假設下，模擬出損失資料，利用最大概似法和動差法來估計此混合分配之參數，並以估計之結果計算出尾部純保費及其信賴區間，以評估尾部純保費的訂價風險。最後，我們也探討了所建議之模式是否可以用來代替其他的中、長尾類型的分配，以檢驗文中建議之模式是否穩健。結果發現，混合 EXP 與 GPD 分配模型的配適效果的確改善傳統模型以及 GPD 模型的問題，而此混合分配將可做為保險含有巨額損失在估算尾部純保費時不錯的選擇。
關鍵字：廣義柏拉圖分配、門檻值、最大概似法、動差法、區間估計、拔靴法

Estimating the Tailed-Loss Net Premiums of Mixed Long-Tailed Distributions

　　　　　　　　　　　　　　　　　　　　　　(Fanny L. F. Lin)＊(Wenhan Hwang)

The proper fitting of loss distributions and the accuracy of parameter estimation are the keys to pricing tailed-loss net premiums. Traditionally, lognormal and Pareto distributions have been used to model this commonly long-tailed, right-skewed data. However, tail probabilities were overestimated or underestimated to the extent that the tailed-loss net premium pricing risk increased. McNeil (1977) and Resnick (1977) used the Generalized Pareto Distribution (GPD) to model data with extreme losses. In this paper we assume a mixture model of the exponential distribution (EXP) and the GPD, simulate losses from it, and then use moment and maximum likelihood methods to estimate parameters. We use fitting and estimation results to calculate confidence intervals and evaluate pricing risks of tailed-loss net premiums. Finally, the robustness of the proposed mixture EXP-GPD model is discussed. In particular, we examine whether EXP-GPD can be used as a substitute for other medium-tailed or long-tailed distributions. As expected, the proposed distributions tail-fitting performance improves on the weaknesses of using a single distribution or the GPD model. The mixed distribution is an appropriate model for data with extreme losses and inference concerning tailed-loss net premiums.
Keywords: generalized pareto distribution, threshold, maximum likelihood estimation, moment method, interval estimation, bootstrap