Oldest-Old Mortality Rates and the Gompertz Law: A Theoretical and Empirical Study Based on Four Countries

Abstract

高齡老人(65歲以上)的死文率歷來都被認為滿足Gompertz模型，實證中有不少高齡資料以此假設來配置，但因受限於資料量的不足，鮮少有研究以統計檢定的觀點檢視Gompertzai模型是否合適，因而至今仍有正反兩派意見支持與反對將老年人死亡率以Gompertz模型來配適或推估。有鑑於此，本研究提出一套檢測Gompertz假設的方法，包括模型的檢定、參數的估計、及藉由電腦模擬(拔靴法)得到參數估計值的標準差。值得一提的是本方法也可用於僅有單一年度各年齡別的死亡資料。有別於過去研究中統計檢定需要兩組以上(也就是兩個以上的年度)的各年齡別的死亡資料。本文除了實證部分採用日本、瑞典、法國、美國這四個國家80歲以上的死亡資料(資料來源：美國柏克萊大學死亡率資料庫與Kannisto, 1994)，也藉由電腦模擬檢驗本文提出的方法是否正確。除此之外，實驗中常因樣本數的不足，而將原先單一年度的資料延長成合併數個年度(通常為五年或十年)的資料，本文第二部分也將探討這種將不同特性的樣本合併對檢測Gompertz假設可能造成的影響。

Testing the Gompertz law (i.e. the law of geometrical progression) for elderly mortality rates has long been discussed in the literature, but tests based on a set of yearly age-specific data have not been fully explored yet. In the first part of this paper, we propose a standard operating procedure for testing Gompertz assumption using yearly age-specific mortality data. Methods used in the procedure include estimation of parameters in the Gompertz law and their standard errors via bootstrap simulation. In addition to the oldest-old (i.e. ages 80 and above) data from Japan, Sweden, France, and the U.S. (Data sources: Berkeley Mortality Database and Kannisto, 1994), a simulation study is used to demonstrate the validity of the proposed procedure. In practice the period of data collection is often prolonged to 5 or 10 years in order to accumulate sufficient sample sizes. However, a longer data collection period is likely to mix data with different attributes and cause problems in the parameters estimation. Thus, in the second part of this appear, we discuss the impacts of the data collection period and population sizes on the testing results.