Taylor's law is a widely observed empirical pattern that relates the variances to the means of population densities. We present four extensions of the classical Taylor's law (TL): (1) a cubic extension of the linear TL describes the mean–variance relationship of human mortality at subnational levels well; (2) in a time series, long-run variance measures not only variance but also autocovariance, and it is a more suitable measure than variance alone to capture temporal/spatial correlation; (3) an extension of the classical equally weighted spatial variance takes account of synchrony and proximity; (4) robust linear regression estimators of TL parameters reduce vulnerability to outliers. Applying the proposed methods to age-specific Japanese subnational death rates from 1975 to 2018, we study temporal and spatial variations, compare different coefficient estimators, and interpret the implications. We apply a clustering algorithm to the estimated TL coefficients and find that cluster memberships are strongly related to prefectural gross domestic product. The time series of spatial TL coefficients has a decreasing trend that confirms the narrowing gap between rural and urban mortality in Japan.

中文翻译：

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时间和空间泰勒定律：在日本地方死亡率中的应用

泰勒定律是一种广泛观察到的经验模式，它将方差与人口密度的均值联系起来。我们提出了经典泰勒定律 (TL) 的四个扩展：(1) 线性 TL 的三次扩展很好地描述了地方层面人类死亡率的均值-方差关系；(2) 在时间序列中，长期方差不仅可以衡量方差，还可以衡量自协方差，它比单独的方差更适合捕捉时间/空间相关性；(3) 经典等权重空间方差的扩展考虑了同步性和邻近性；(4) TL 参数的稳健线性回归估计器减少了对异常值的脆弱性。将所提出的方法应用于 1975 年至 2018 年特定年龄的日本地方死亡率，我们研究了时空变化，比较不同的系数估计量，并解释其含义。我们将聚类算法应用于估计的 TL 系数，发现聚类成员与县级国内生产总值密切相关。空间 TL 系数的时间序列呈下降趋势，证实了日本农村和城市死亡率之间的差距正在缩小。