高龄老人死亡率与 Gompertz 模型理论与实证研究

中文字数:3072, 英文字数:5340

高龄老人死亡率与 Gompertz 模型理论与实证研究
摘    要
 高龄老年人(65岁以上)的死亡率历来都被认为满足 Gompertz 模型，实证中有不少高龄资料以此假设来配适，但因受限于资料量的不足，鲜少有研究以统计检定的观点检视Gompertz 模型是否合适，因而至今仍有正反两派意见支持与反对将老年人死亡率以Gompertz 模型来配适或推估。有鉴于此，本研究提出一套检测 Gompertz 假设的方法，包括模型的检定、参数的估计、及藉由计算机仿真(拔靴法)得到参数估计值的标准差。值得一提的是本方法也可用于仅有单一年度各年龄别的死亡资料，有别于过去研究中统计检定需要两组以上(也就是两个以上的年度)的各年龄别的死亡资料。本文除了实证部分采用日本、瑞典、法国、美国这四个国家80岁以上的死亡数据(数据来源：美国柏克莱大学死亡率数据库与 Kannisto, 1994)，也藉由计算机仿真检验本文提出的方法是否正确。除此之外，实证中常因样本数的不足，而将原先单一年度的资料延长成合并数个年度(通常为五年或十年)的数据，本文第二部分也将探讨这种将不同特性的样本合并对检测 Gompertz 假设可能造成的影响。
关键词： Gompertz 模型、高龄死亡率、计算机仿真、最大概似估计、加权最小平方法

Oldest-Old Mortality Rates and the Gompertz Law:
A Theoretical and Empirical Study Based on Four Countries
Jack C. Yue
Abstract
 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 the 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 parameter estimation. Thus, in the second part of this paper, we discuss the impacts of the data collection period and population sizes on the testing results.
 Key Words: Gompertz’s law,　mortality rates of the elderly,　simulation,　maximum likelihood, weighted least square

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