Multiplicative Decomposition of Heterogeneity in Mixtures of Continuous Distributions

Abstract

A system's heterogeneity (\textit{diversity}) is the effective size of its event space, and can be quantified using the R\'enyi 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 $\gamma$-heterogeneity) is decomposable into heterogeneity arising from variation \textit{within} and \textit{between} component subsystems (the $\alpha$- and $\beta$-heterogeneity, respectively). Since the average heterogeneity of a component subsystem should not be greater than that of the pooled system, we require that $\gamma \geq \alpha$. There exists a multiplicative decomposition for R\'enyi 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 R\'enyi heterogeneity for continuous mixture distributions under parametric and non-parametric pooling assumptions. Under non-parametric pooling, the $\gamma$-heterogeneity must often be estimated numerically, but the multiplicative decomposition holds such that $\gamma \geq \alpha$ for $q > 0$. Conversely, under parametric pooling, $\gamma$-heterogeneity can be computed efficiently in closed-form, but the $\gamma \geq \alpha$ condition holds reliably only at $q=1$. Our findings will further contribute to heterogeneity measurement in continuous systems.