混合连续分布下异方差的乘性分解
Multiplicative Decomposition of Heterogeneity in Mixtures of Continuous Distributions
摘要：
系统的异质性（\textit{diversity}）是其事件空间的有效大小，可以使用雷尼指数族（也称为生态学中的希尔数或经济学中的汉纳-凯指数）进行量化，这些指数由弹性参数$q \geq 0$索引。在这些指标下，复合系统的异质性（$γ$ -异质性）可分解为由\textit{within}和\textit{between}组成子系统的变化引起的异质性（分别为$α$ -异质性和$β$ -异质性）。由于组件子系统的平均异质性不应大于合并系统的异质性，因此我们要求$γ\geq α$。离散事件空间下的复合系统的Rényi非齐性存在乘性分解，而连续事件空间下的分解研究较少.因此，我们描述了乘法分解的Rényi异质性连续混合分布参数和非参数池假设。在非参数合并下，$γ$ -异质性通常必须用数值估计，但乘法分解保持，使得$q > 0$的$γ\geq α$。相反，在参数合并下，$γ$ -异质性可以以封闭形式有效地计算，但是$γ\geq α$条件仅在$q=1$处可靠地保持。我们的研究结果将进一步有助于连续系统的异质性测量。
英文摘要：
A system's heterogeneity (\textit{diversity}) is the effective size of its event space, and can be quantified using the Rényi family of indices (also known as Hill numbers in ecology or Hannah-Kay indices in economics), which are indexed by an elasticity parameter $q \geq 0$. Under these indices, the heterogeneity of a composite system (the $γ$-heterogeneity) is decomposable into heterogeneity arising from variation \textit{within} and \textit{between} component subsystems (the $α$- and $β$-heterogeneity, respectively). Since the average heterogeneity of a component subsystem should not be greater than that of the pooled system, we require that $γ\geq α$. There exists a multiplicative decomposition for Rényi heterogeneity of composite systems with discrete event spaces, but less attention has been paid to decomposition in the continuous setting. We therefore describe multiplicative decomposition of the Rényi heterogeneity for continuous mixture distributions under parametric and non-parametric pooling assumptions. Under non-parametric pooling, the $γ$-heterogeneity must often be estimated numerically, but the multiplicative decomposition holds such that $γ\geq α$ for $q > 0$. Conversely, under parametric pooling, $γ$-heterogeneity can be computed efficiently in closed-form, but the $γ\geq α$ condition holds reliably only at $q=1$. Our findings will further contribute to heterogeneity measurement in continuous systems.
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