Currie Durban and Eilers 2004 — Smoothing and Forecasting Mortality Rates

Summary

Currie, Durbán, and Eilers propose fitting the two-dimensional surface of log-mortality rates using penalized B-splines (P-splines) within a Poisson generalized linear model (GLM), replacing the Lee-Carter (LC) singular value decomposition (SVD) with a unified smoothing-and-forecasting framework. Deaths $d_{xt}$ are Poisson with mean $e_{xt}\mu_{xt}$; the log-mortality surface $\log\mu_{xt}$ is modeled as a tensor-product B-spline with separate second-order difference penalties in the age and year dimensions. Estimation via Generalized Linear Array Models (GLAMs) and iteratively reweighted least squares (IRLS) is computationally efficient, produces a smooth mortality surface, and extrapolates naturally in the year direction to produce forecasts with approximate posterior prediction intervals. Cohort effects can be captured by adding diagonal P-splines.

Key Claims

• Model: $d_{xt} \sim \text{Poisson}(e_{xt}\mu_{xt})$; $\log\mu_{xt} = \mathbf{B}_x \boldsymbol{\Theta} \mathbf{B}_t^\top$ where $\mathbf{B}_x$ and $\mathbf{B}_t$ are B-spline bases for age and year, and $\boldsymbol{\Theta}$ is a coefficient matrix penalized by $\lambda_x P_x + \lambda_t P_t$ (second-order difference penalties in age and year respectively).
• GLAM estimation: The Kronecker product structure of the tensor-product basis allows computation on coefficient arrays rather than full vectorized design matrices, reducing memory and runtime by orders of magnitude for large mortality tables.
• Unified framework: Smoothing and forecasting are one operation — extending $\mathbf{B}_t$ to future years extrapolates the fitted surface; no separate time-series model for a latent index is required.
• Uncertainty: Prediction intervals derive from the approximate posterior distribution of the coefficient matrix $\boldsymbol{\Theta}$ at the IRLS solution, without a separate stochastic mortality model.
• Cohort effects: Diagonal P-splines (penalties along age-year diagonals representing birth cohorts) capture cohort structure — demonstrated on the UK "golden cohort" (born ca. 1925–1935) whose unusually rapid mortality improvement is invisible to age-period models.
• Comparison to Lee-Carter: Lee-Carter imposes a rank-1 separable bilinear structure $a_x + b_x k_t$; P-splines fit a fully flexible smooth surface with no separability constraint, nesting LC as a special case.
• Application: Demonstrated on England and Wales male mortality 1968–1997; forecasts to 2017 compare favorably to Lee-Carter and show the cohort P-spline capturing future improvement patterns that period models miss.
• Smoothing parameter selection: Akaike Information Criterion (AIC) or cross-validation selects $\lambda_x$ and $\lambda_t$; results are robust to the choice over a broad range.

Concepts Introduced or Extended

• P-spline Mortality Smoothing — the central methodological contribution; 2D P-splines on the Poisson log-mortality surface, unifying smoothing and forecasting in a single GLM

Entities Mentioned

• Iain D. Currie
• María Durbán
• Paul H.C. Eilers

Quotes

"The key idea is to use P-splines to smooth and forecast the surface of mortality rates in a single operation."

"The Lee-Carter model can be seen as a low-rank approximation to the log mortality surface. Our approach makes no such constraint, fitting a fully flexible smooth surface."

My Take

The paper's central contribution is methodological elegance: replacing Lee-Carter's two-stage SVD + autoregressive integrated moving average (ARIMA) pipeline with a single Poisson GLM that simultaneously smooths and extrapolates. The GLAM framework makes this computationally tractable for mortality arrays of practical size. The cohort extension is practically important for countries (notably the UK) where cohort effects are strong enough to dominate period-by-period projections — something the standard Lee-Carter framework cannot represent without extension. The main limitation is that the forecast rests on smoothness assumptions with no structural interpretation: the implicit model for future mortality is "approximately polynomial," which lacks the formal Brownian motion interpretation of Lee-Carter's random walk with drift. The paper laid the groundwork for the Renshaw-Haberman (2006) age-period-cohort (APC) extension and the broader family of GLM-based mortality models, and P-splines are now a standard tool in the StMoMo R package used by UK and European actuaries for Solvency II reserve calculations.