Multiplicative Decomposition of Heterogeneity in Mixtures of Continuous Distributions

A system’s heterogeneity (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 [Formula: see text]. Under these indices, the heterogeneity of a composite system (the [Formula: see text]-heterogeneity) is decomposable into heterogeneity arising from variation within and between component subsystems (the [Formula: see text]- and [Formula: see text]-heterogeneity, respectively). Since the average heterogeneity of a component subsystem should not be greater than that of the pooled system, we require that [Formula: see text]. 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 [Formula: see text]-heterogeneity must often be estimated numerically, but the multiplicative decomposition holds such that [Formula: see text] for [Formula: see text]. Conversely, under parametric pooling, [Formula: see text]-heterogeneity can be computed efficiently in closed-form, but the [Formula: see text] condition holds reliably only at [Formula: see text]. Our findings will further contribute to heterogeneity measurement in continuous systems.