Structured dispersion modelling for mortality data using a Conway--Maxwell--Poisson specification

Technical Summary: Structured Dispersion Modelling for Mortality Data using a Conway–Maxwell–Poisson Specification

Problem Statement
Traditional mortality forecasting models, such as the Lee–Carter (LC) and Cairns–Blake–Dowd (CBD) families, typically assume a Poisson distribution for death counts. This assumption imposes mean-variance equality ($E[D_{xt}] = \text{Var}[D_{xt}]$), which frequently fails to hold in empirical mortality data due to unobserved heterogeneity. While existing literature has addressed overdispersion through global dispersion parameters (e.g., quasi-Poisson, negative-binomial, or single global Conway–Maxwell–Poisson specifications), these approaches often treat dispersion as a fixed, aggregate quantity. The authors argue that this assumption is rarely satisfied in practice; dispersion is likely a structured quantity that varies systematically across age and time. Failing to capture this structural heterogeneity can lead to miscalibrated uncertainty, overly narrow prediction intervals, and an underestimation of longevity risk.

Methodology
The paper proposes a flexible framework that explicitly models dispersion as a function of age and time using the Conway–Maxwell–Poisson (CMP) distribution.

1. Distributional Specification:
   The core innovation is moving from a single global dispersion parameter ($\nu$) to age-specific ($\nu_x$) and period-specific ($\nu_t$) parameters. The CMP distribution allows for underdispersion ($\nu > 1$), equidispersion ($\nu = 1$), and overdispersion ($\nu < 1$).

   • Age-specific models (CMP–LC–x, CMP–LCC–x): The dispersion parameter varies by age group $x$.
   • Period-specific models (CMP–LC–t, CMP–LCC–t): The dispersion parameter varies by calendar year $t$.
   • These are combined with standard mortality rate structures: the Lee–Carter (LC) model and the Lee–Carter with Cohort (LCC) model.

2. Bayesian Formulation:
   The authors adopt a Bayesian framework to treat dispersion parameters as unknown random variables.

   • Priors: Vague Gamma priors ($\text{Gamma}(1, 0.01)$) are assigned to $\nu_x$ and $\nu_t$ to allow data-driven inference without pre-specifying the dispersion structure. Standard priors are used for mortality rate parameters ($\alpha_x, \beta_x, \kappa_t, \gamma_c$), including ARIMA-type priors for time and cohort trends.
   • Time-Series Extension: For period-specific dispersion, the paper explores an ARIMA(0,1,1) prior with linear drift to capture temporal correlations and trends in the dispersion process, allowing for dynamic projection of heterogeneity.
   • Estimation: Parameters are estimated using Markov Chain Monte Carlo (MCMC) methods. The updating scheme for dispersion parameters involves log-scale Metropolis-Hastings steps to ensure positivity.

Key Contributions
The paper identifies three primary contributions:

1. Flexible Dispersion Structure: Unlike previous CMP applications that use a single global $\nu$, this framework allows the type and degree of dispersion to vary by age and time. This enables the inference of underlying dispersion structures directly from the data without pre-specification.
2. Coherent Bayesian Implementation: The authors present a Bayesian implementation that integrates parameter, process, and distributional uncertainty. This avoids the two-stage estimation procedures common in frequentist approaches and utilizes prior structures to propagate uncertainty coherently.
3. Empirical Validation and Actuarial Application: The methodology is applied to male death data from England and Wales (1961–2002). The study demonstrates that variability in mortality counts differs substantially across ages and periods, with implications for the calibration of longevity risk and annuity pricing.

Results
Using England and Wales male mortality data, the study compares Poisson, standard CMP, and structured CMP models (both LC and LCC variants).

• Dispersion Patterns: Empirical analysis reveals that dispersion is not uniform. Significant overdispersion is observed at infant ages, ages 18–23, 45–50, 60–65, and 80–95. Period-specific analysis shows strong overdispersion in years such as 1963, 1969, 1985, 1994, and 2000–2002.
• Model Fit: Structured CMP models (specifically CMP–LCC–x and CMP–LCC–t) significantly outperform Poisson and global CMP models.
  • Residuals: The CMP–LCC–x model yields the lowest total residuals for death counts, effectively alleviating clusters of discrepancies found in standard models (e.g., at infant ages and ages 18–23).
  • Annuity Pricing: In terms of annuity-due values, the CMP–LCC–t model performs best, providing better calibration of interval widths.
• Uncertainty Calibration: Models with structured dispersion produce more realistic prediction intervals. Global models tend to produce overly wide intervals for some ages (due to averaging) and overly narrow intervals for others. Structured models adapt interval widths to local data characteristics, improving the coverage of holdout data.
• Forecasting: The time-series prior for $\nu_t$ (ARIMA(0,1,1)) successfully captures a slowly evolving stochastic trend, suggesting that the level of overdispersion may diminish gradually in the future, though the authors note limitations regarding potential negative projections and indefinite linear trends.

Significance and Claims
The paper claims that explicitly modelling structural dispersion offers a superior alternative to models relying on a single global dispersion parameter. By inferring the dispersion structure directly from the data, the proposed framework:

• Captures residual heterogeneity that standard overdispersion models overlook.
• Provides a more localized calibration of uncertainty, which is critical for accurate longevity risk assessment and annuity pricing.
• Demonstrates that the degree of stochastic variation in mortality is not constant but varies systematically across the age spectrum and over time.

The authors conclude that combining flexible distributional assumptions (CMP) with structural dispersion components within a coherent Bayesian framework delivers improved model fit and a richer representation of mortality heterogeneity. They modestly suggest that future work could explore formal model selection for time-series specifications and alternative priors that ensure non-negativity and long-term stability in dispersion projections.