Kim Y J
Demography. 1986 Aug;23(3):451-65.
The formula for the age distribution and other relationships that follow from it for any (non-stable) population presented by Preston and Coale are significant contributions to demography. The formulas summarize the relationships among various demographic measures precisely, and are formally analogous to the relationships that hold for stable populations. The significance of these formulas cannot be overstated; they allow us to understand clearly the relationships among demographic measures in any arbitrary population. However, when it comes to using them for estimating demographic measures when census data are defective, the method of estimation is still affected by defective data. The reason is that the series of age-specific growth rates reflects the observed census age distributions exactly so that any defects in the census data are summarized in the growth rates. This paper begins with the formulation of the discrete version of the "new synthesis" developed by Preston and Coale. With the discrete formulation, the three kinds of errors introduced when the continuous time formulas are applied to real data can be avoided. Then it is pointed out that when two accurate census data are available, the Preston-Coale procedure of "estimating" the age distribution at the second census is equivalent to checking the identity of the age distribution formula. Also "estimating" mortality by the procedure of Preston-Coale is shown to be equivalent to obtaining mortality directly from intercensal survival rates. That the procedure which involves the age-specific growth rates is equivalent to those that involve the intercensal survival rates may have escaped notice because there are no a priori constraints for patterns of age-specific growth rates to follow. The irregularity in growth rates due to defective data are not distinguishable from true irregularity that exists in the population, contrary to the well-known regularity in the pattern of survival rates in human populations.
普雷斯顿和科尔提出的关于任何（非稳定）人口年龄分布及其衍生的其他关系的公式，是对人口统计学的重大贡献。这些公式精确地总结了各种人口统计指标之间的关系，并且在形式上类似于稳定人口所具有的关系。这些公式的重要性怎么强调都不为过；它们使我们能够清楚地理解任意人口中人口统计指标之间的关系。然而，当在人口普查数据存在缺陷时使用这些公式来估计人口统计指标时，估计方法仍然会受到有缺陷数据的影响。原因是特定年龄增长率序列精确反映了观察到的人口普查年龄分布，以至于人口普查数据中的任何缺陷都体现在增长率中。本文首先阐述了普雷斯顿和科尔所提出的“新综合”的离散形式。通过离散形式，可以避免将连续时间公式应用于实际数据时引入的三种误差。然后指出，当有两个准确的人口普查数据时，普雷斯顿 - 科尔在第二次人口普查时“估计”年龄分布的过程等同于检验年龄分布公式的一致性。同时还表明，通过普雷斯顿 - 科尔程序“估计”死亡率等同于直接从两次人口普查间的生存率中获取死亡率。涉及特定年龄增长率的程序等同于涉及两次人口普查间生存率的程序，这一点可能未被注意到，因为对于特定年龄增长率模式没有先验的约束条件。与人类人口中众所周知的生存率模式的规律性相反，由于数据缺陷导致的增长率不规则性与人口中存在的真实不规则性无法区分。
