醫療技術的進步以及生活品質的提升，預計人類平均壽命將持續延長，以臺灣為例，男、女性平均壽命將從2011年的75.98歲、82.65歲，增加到2060年的82.0歲、88.0歲（資料來源：行政院經濟建設委員會2012年推估）。壽命延長意謂更長的退休生活，世界各國在21世紀均面對需求日殷的老年生活照顧，包括退休金制度以及老人醫療等，這些社會福利及保險勢必增加國家財務負擔，因此壽命是否繼續延長或存有極限成為大家關心的議題。近年來，不少研究透過死亡壓縮(Mortality Compression)連結壽命議題，亦即探討死亡年齡是否將集中至更窄的範圍，但因為資料及研究方法的限制，死亡壓縮是否成立仍無定論。 本研究以統計方法、分配假設、資料品質，三個面向來探討死亡壓縮與延壽之間的關係。本研究提出三種數值優化方法：加權最小平方法(Weighted Least Squares；WLS)、非線性極值法(Nonlinear-Maximization；NM)及最大概似估計法(Maximal Likelihood Estimation；MLE)，透過電腦模擬衡量方法優劣，與過去常見的方法比較（Kannisto的SD(M+)），探討何者具有較小的均方誤差(Mean Squared Error；MSE)。其次若死亡年齡之真實死亡分配為t分配時，探討以常態假設代入計算所產生的偏誤；最後則是套入各國實際死亡資料，使用上述較佳的估計方法，檢視死亡壓縮是否存在。 研究結果顯示，NM具有不偏性質且具有較小的均方誤差，過去研究常用的SD(M+)反而有明顯偏誤，且隨著觀察值越多變異數反而增加。而若真實死亡分配若為t分配時，以原先利用常態假設所計算的年金險保費皆有低估的情形，分配的重要性可見一斑，進而探討在實務上常態分配之假設，發現與仍與實際情形有明顯之差異，不論是NM及SD(M+)在死亡壓縮的探討下，皆受到資料的限制而有待商榷。 / Due to the advance in medical technology and the change of life style, the human life expectancy has been increasing since the end of the Second World War II and it is expected to continue the pace of increment. Longer life expectancy also means a longer life after retirement. People living in the 21st century are faced with growing demand for the retirement life, such as the pension funds and medical needs to the individuals, as well as the social welfare and insurance for the elderly to the government. Thus, the issue whether the lifespan has a limit receives a lot of attention. In particular, many studies focus on the topic of mortality compression, which means that the expectancy of lifespan has a limit and variance of lifespan converge. However, due to the availability of elderly data, there is still no consensus if the mortality compression is true. In this study, we propose estimation methods to estimate modal age and variance of the age-at-death. Three types of methods are involved: weighted least squares (WLS) method, nonlinear maximization (NM) method, and maximum likelihood estimation (MLE) method, and they are compared to the method proposed by Kannisto, namely SD(M+), in 2000. We found that the NM method has a smaller MSE, and we cannot decide the mortality compression is true based on the data from Human Mortality Database. We also applied the normality and t distribution assumption to the age-at-death and compute the pure premiums for annuity products. We found that normality distribution would produce larger premiums than using the empirical mortality rates. Similarity, the bankruptcy probability would be higher if the t distribution is used.

死亡壓縮(Mortality Compression)意指死亡年齡更集中，是最近廣受注意的研究議題，和生存曲線矩形化（Rectangularization）關係密切，以統計分佈的角度描述，則是死亡年齡會逐漸退化到某個特定年齡。換言之，如果死亡壓縮和壽命有上限兩者都成立，以統計術語而言，代表壽命的期望值有上限、變異數會收斂，可藉由死亡年齡分配探討壽命變化。 本文希望以統計方法與資料品質等兩個面向探討死亡壓縮與延壽之間的關係。除了過去使用的無母數方法，如檢視各年度生命表上死亡分佈的最短區間（25%、50%及75%）與死亡人數最多的年齡（Modal Age）的變化，探討死亡壓縮與壽命是否有延長；另一方面，也將對死亡曲線作參數設定，觀察死亡年齡分佈的標準差變化。由於過往的研究多使用的生命表資料，本研究將比較使用生命表資料（死亡資料經過修勻）或原始死亡人數資料對結果的影響。 本研究藉由電腦模擬比較各種估計標準差方法的差異，包括Kannisto (2000) 提出的SD(M+)法與本文考量的非線性極值法(Nonlinear-Maximization)，衡量何者具有較小的均方誤差，並探討錯誤設定分配偵誤的敏感度；另外，本文可討論使用經過修勻的死亡率及原始死亡率對於估計結果的影響。除了電腦模擬，本研究也套入實際死亡資料（如臺灣、美國、…等國資料，資料來源：Human Mortality Database），檢視死亡壓縮是否存在。 / Mortality compression is one of the popular research issues in longevity risk. It means that the age-at-death would concentrate on a narrower range, and it is also related to the concept of rectangularization of survival curve. In terms of statistical distribution, mortality compression indicates that the age-at-death degenerates to a certain age, and it can be used to study changes of lifespan. If the lifespan has a limit, or mortality compression does exist, this suggests that the life expectancy has a limit and the variance of age-at-death would converge. In the study, we evaluate the mortality compression using the statistical methods and considering the issue of data quality. In addition to the nonparametric methods used in the previous studies, such as shortest confidence interval on the distribution of age-at-death and the modal age, we consider optimization methods for estimating the standard deviation of age-at-death distribution. In specific, we compare the SD(M+) proposed by Kannisto (2000) and the method of Nonlinear-Maximization, and check which method has a smaller MSE (Mean Squared Error). For the issue of data quality, we compare the estimation results of using mortality rates from life table data with those using the raw data. In addition to computer simulation, we consider the sensitivity analysis of age-at-death distribution, to evaluate the estimation method. Furthermore, based on the data from Human Mortality Database, we apply the method of Nonlinear-Maximization to life table data (i.e., graduated mortality rates) and raw data, and check if there are significant differences. The estimation results of empirical study are also used to evaluate if there is mortality compression and if there is a longevity limit.