Life Table Functions and Their Uncertainty: A Matrix Algebra Approach

Carlo-Giovanni Camarda , Ined

The construction of confidence intervals for life expectancy at birth has a long tradition in demography, reflecting the central role of uncertainty in mortality estimation, particularly when analyzing small populations or subnational areas. Yet, despite this longstanding interest, no unified statistical framework currently exists for quantifying uncertainty across all life table functions. This paper introduces a coherent, statistically grounded, and computationally refined approach for propagating uncertainty throughout the life table. Building on the Poisson nature of death counts, we derive and compare three complementary methods for quantifying uncertainty in age-specific mortality: the Delta method, the Garwood exact approach, and a Bayesian bootstrap based on Poisson-Gamma conjugacy. These methods are then embedded in a matrix algebra formulation that unifies the computation of all major life table functions, survivorship, deaths, person-years, total years remaining, and life expectancy, and their associated variance–covariance structures. All analytical derivations are presented explicitly, providing both a methodological framework for uncertainty propagation and a formal, elegant reinterpretation of the life table as an interconnected stochastic system. The approach is illustrated using 2020 female mortality data from the 20 administrative districts (arrondissements) of Paris, highlighting how uncertainty behaves across populations of varying size and mortality levels. Finally, several extensions are outlined, including the quantification of uncertainty for lifespan variability measures, the adaptation to abridged life tables, and applications to model-based and comparative mortality analyses. These developments lay the groundwork for a unified, probabilistic framework for the study of demographic functions.

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 Presented in Session 39. Innovations in Life Table and Multistate Modeling