Distributional Frailty Mixtures for Overdispersed Count Regression

Abstract

Standard finite mixtures of negative binomial regressions allow covariate-dependent means within latent classes but impose constant dispersion within each component. We propose the distributional finite exponential–gamma frailty mixture (D-FEGFM) regression model, which allows component-specific dispersion to depend on covariates through a log-linear regression. The model nests the constant-dispersion mixture and single-component distributional negative binomial regression. We derive the likelihood, establish identifiability up to label switching, extend the within/between variance decomposition to covariate-varying dispersion, and develop a generalized EM algorithm. A Monte Carlo study shows that the model is identifiable but weakly powered when mean and dispersion share the same covariates. In healthcare utilisation data, the strongest gains arise when demographic variables drive dispersion and clinical variables drive the mean, improving fit and held-out predictive performance, and yielding an interpretable insurance effect on utilisation regularity.