A system's heterogeneity ( equivalently, diversity) amounts to 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. Importantly, under these indices, the heterogeneity of a composite system (the \gamma-heterogeneity) is decomposable into heterogeneity arising from variation within and between component subsystems (the \alpha- and \beta -heterogeneities , respectively). Since the average heterogeneity of a component subsystem should not be greater than that of the pooled assemblage , we require that \alpha \leq \gamma . 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. This paper therefore describes multiplicative decomposition of the R\'enyi heterogeneity for continuous mixture distributions under parametric and non-parametric pooling assumptions. We show that under non-parametric pooling (where \gamma-heterogeneity must typically be estimated numerically ), the multiplicative decomposition holds such that \gamma \geq \alpha for all values of the elasticity parameter q > 0, and \beta-heterogeneity amounts to the discrete number of distinct mixture components in the system. Conversely, under parametric pooling (as in a Gaussian mixed-effects model), which facilitates efficient analytical computation of \gamma-heterogeneity , we show that the \gamma \geq \alpha condition holds  only at q=1. By providing conditions under which the decomposability axiom of heterogeneity measurement holds, our findings further advance the understanding of heterogeneity measurement in non-categorical systems.
A system's heterogeneity ( 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 within and 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.
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