Age-Specific Mortality in Italian Provinces via Functional Generalized Linear Mixed Models
M. Scianna${}^a$ and C. Agostinelli${}^a$
${}^a$Department of Mathematics, University of Trento
Generalized Linear Mixed Models (GLMMs) are a standard tool for longitudinal and grouped data, yet their classical formulation assumes scalar predictors, discarding the granular information carried by covariates that are more naturally represented as functions. This work addresses a primary theoretical gap: the incorporation of distribution functions as functional covariates within a FGLMM framework. Treating predictors as distributions – rather than general smooth functions – allows one to assess how shifts in the mass of a population profile propagate into a scalar response, a setting not covered by existing scalar-on-function regression theory. The proposed approach combines B-spline basis expansions with functional principal component analysis (FPCA) to reduce the infinite-dimensional covariate to a finite vector of uncorrelated scores [1]. These scores enter a hierarchical mixed-effects model with specific random intercepts and functional coefficients, estimated via both a Bayesian Hamiltonian Monte Carlo scheme [4] and a frequentist maximum likelihood baseline. The close agreement between the two provides a robustness check on the inferred functional effects. The framework is validated on ISTAT data covering 107 Italian provinces over 2011–2023, where age distributions serve as functional covariates and crude mortality rates as the response. Temporal regimes are identified non-parametrically through pairwise stochastic dominance tests [2], avoiding parametric assumptions on the response distribution. Reconstructed coefficients confirm a stable pre-pandemic age–mortality gradient, a marked amplification of middle-age contributions in 2020 consistent with the demographic profile of COVID-19 [3], and a progressive return to baseline through 2023. The application demonstrates that distributional covariates can be embedded in mixed models in a computationally feasible and interpretable way. Keywords: Functional data analysis, Mortality modelling, Mixed models.