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arxiv: 2507.12330 · v2 · submitted 2025-07-16 · 📊 stat.AP · stat.ME

Forecasting sub-population mortality using credibility theory

Mathias Lindholm , Gabriele Pittarello This is my paper

Pith reviewed 2026-05-19 04:06 UTC · model grok-4.3

classification 📊 stat.AP stat.ME
keywords mortality forecastingcredibility theorysub-populationsuper-populationforecast errorlatent stochastic processLee-Carter models
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The pith

Credibility theory produces weighted-average forecasts for small sub-population mortality rates by blending super-population and local predictions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish a way to forecast mortality rates for small sub-populations by extending credibility theory to handle latent stochastic processes like those in Lee-Carter models. It derives predictors that are weighted averages between the super-population forecast and the sub-population forecast, along with an explicit formula for the expected squared forecast error. This matters because many real-world groups, such as specific cities or demographic slices, have too little data for standalone models yet are part of larger populations with good data. If correct, it provides a flexible compromise that improves accuracy without needing to overhaul the super-population model.

Core claim

The paper claims that when future mortality rates follow a latent stochastic process, the credibility predictor for a sub-population is a weighted average of the expected super-population mortality and the expected sub-population mortality, and that an explicit expression for the expected quadratic forecast error can be derived. The approach remains valid irrespective of the exact form chosen for the super-population forecasting model.

What carries the argument

A linear credibility predictor that computes a weighted average of super-population and sub-population expected mortality rates to balance global reliability against local specificity.

If this is right

  • The credibility predictor reduces to a simple weighted average of super-population and sub-population expectations.
  • The expected quadratic forecast error has a closed-form expression.
  • The method applies regardless of the super-population model chosen.
  • It can be illustrated on simulated data showing it as a compromise between the two extremes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This blending technique may apply to forecasting other quantities in hierarchical populations, such as economic or health metrics.
  • Empirical validation could involve applying the method to historical data from countries with regional breakdowns.
  • Extensions might incorporate time-varying weights if sub-population data accumulates over time.

Load-bearing premise

Reliable forecasts exist for the overall super-population even though individual sub-populations have insufficient data or history for separate modeling, under a latent stochastic process for mortality rates.

What would settle it

Observing whether the mean squared prediction error on future mortality data for small sub-populations is smaller for the credibility predictor than for either the super-population-only or sub-population-only forecasts.

Figures

Figures reproduced from arXiv: 2507.12330 by Gabriele Pittarello, Mathias Lindholm.

Figure 1
Figure 1. Figure 1: Sub-populations 1, and 2 are part of a larger reference super-population. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Credibility model results on a simulated data-set using the Lee-Carter for modelling the mortality trend. [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Test (red) split of the out-of-sample data in the rolling window approach. The in-sample data are in [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The gray dots represent observed central mortality rates for the synthetic data for age [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The three rows show the performance metrics for the super-population, the medium size sub-population [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: On the left-hand side, assumptions on the mortality trend for [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
read the original abstract

The focus of the present paper is to forecast mortality rates for small sub-populations that are parts of a larger super-population. In this setting the assumption is that it is possible to produce reliable forecasts for the super-population, but the sub-populations may be too small or lack sufficient history to produce reliable forecasts if modelled separately. This setup is aligned with the ideas that underpin credibility theory, and in the present paper the classical credibility theory approach is extended to be able to handle the situation where future mortality rates are driven by a latent stochastic process, as is the case for, e.g., Lee-Carter type models. This results in sub-population credibility predictors that are weighted averages of expected future super-population mortality rates and expected future sub-population specific mortality rates. Due to the predictor's simple structure it is possible to derive an explicit expression for the expected quadratic forecast error. Moreover, the proposed credibility modelling approach does not depend on the specific form of the super-population model, making it broadly applicable regardless of the chosen forecasting model for the super-population. The performance of the suggested sub-population credibility predictor is illustrated on simulated population data. These illustrations highlight how the credibility predictor serves as a compromise between only using a super-population model, and only using a potentially unreliable sub-population specific model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript extends classical credibility theory to forecast mortality rates for small sub-populations within a larger super-population. It assumes reliable super-population forecasts are available while sub-populations lack sufficient data or history for standalone modeling. The approach incorporates latent stochastic processes (e.g., Lee-Carter models) to produce sub-population predictors as weighted averages of expected future super-population and sub-population mortality rates. An explicit expression for the expected quadratic forecast error is derived, and the modeling framework is presented as independent of the specific super-population model form. Performance is illustrated via simulations on population data.

Significance. If the central derivations hold, the work supplies a practical, flexible method for improving forecasts in small populations by borrowing strength from larger ones while providing explicit error quantification. The claimed model-agnostic property and simple predictor structure could facilitate adoption across different stochastic mortality models in actuarial and demographic applications. The simulated illustrations offer initial evidence of the compromise between super-population and sub-population modeling.

major comments (2)
  1. [Abstract] Abstract: The central claim that 'the proposed credibility modelling approach does not depend on the specific form of the super-population model' is load-bearing for the paper's generality. However, the explicit quadratic forecast error expression requires the second-moment structure induced by the latent process dynamics (e.g., variance of the time-index forecast under the chosen time-series model). This structure is model-specific, so the independence holds for the algebraic shape of the predictor but not necessarily for the error formula or the resulting numerical weights. A concrete test would be to substitute two different latent-process specifications and verify whether the error expression remains unchanged beyond the predictor form.
  2. [Section 3.2, Equation (8)] Section 3.2, Equation (8): The credibility weights are functions of variance parameters estimated from the same data used to compute the forecast error. The manuscript should clarify whether the derived error expression treats these parameters as fixed or accounts for their estimation uncertainty, as this affects whether the error formula is truly explicit and non-circular.
minor comments (3)
  1. [Section 2.1] Section 2.1: Define the credibility weights explicitly with a table or numbered display rather than inline text to improve readability.
  2. [Figure 2] Figure 2: Add error bars or shaded regions to the simulated forecast comparisons to better visualize variability across replications.
  3. [References] References: Include at least one recent reference on credibility applications in mortality modeling to situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. We address each major comment below, clarifying the scope of our claims and indicating revisions to improve precision without altering the core contributions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that 'the proposed credibility modelling approach does not depend on the specific form of the super-population model' is load-bearing for the paper's generality. However, the explicit quadratic forecast error expression requires the second-moment structure induced by the latent process dynamics (e.g., variance of the time-index forecast under the chosen time-series model). This structure is model-specific, so the independence holds for the algebraic shape of the predictor but not necessarily for the error formula or the resulting numerical weights. A concrete test would be to substitute two different latent-process specifications and verify whether the error expression remains unchanged beyond the predictor form.

    Authors: We agree that the algebraic form of the credibility predictor is independent of the specific super-population model, while the numerical values in the quadratic error expression depend on the second-moment quantities supplied by the chosen latent process. The abstract claim refers specifically to the modeling framework and predictor structure being applicable to any super-population model that provides forecasts and associated variances; the error formula is expressed in terms of these general moments rather than being tied to one particular time-series specification. We will revise the abstract and add a clarifying paragraph in Section 2 to distinguish the model-agnostic predictor from the model-dependent inputs to the error formula. A direct substitution test is not required for the derivation but could be noted as an illustrative extension. revision: partial

  2. Referee: [Section 3.2, Equation (8)] Section 3.2, Equation (8): The credibility weights are functions of variance parameters estimated from the same data used to compute the forecast error. The manuscript should clarify whether the derived error expression treats these parameters as fixed or accounts for their estimation uncertainty, as this affects whether the error formula is truly explicit and non-circular.

    Authors: The derivation in Section 3.2 treats the variance parameters as fixed once estimated from the data, consistent with the classical credibility approach where parameters are plugged in after separate estimation. The explicit quadratic error expression is therefore conditional on these estimates and does not incorporate their sampling variability. We will add a clarifying sentence in Section 3.2 stating this assumption explicitly and noting that full propagation of estimation uncertainty would require additional techniques such as bootstrap or asymptotic expansions, which lie outside the current scope. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained.

full rationale

The paper extends classical credibility theory to a latent stochastic process setting for sub-population mortality forecasting. Sub-population predictors are constructed as weighted averages of super-population and sub-population expected rates, with an explicit quadratic forecast error derived directly from the algebraic structure of that predictor. The claim of independence from the specific super-population model form follows from the general credibility weighting not requiring insertion of particular time-series dynamics beyond the latent-process assumption. No quoted step reduces a prediction or first-principles result to its own inputs by construction, no fitted parameter is renamed as an independent prediction, and no self-citation chain is load-bearing for the central claims. The derivation remains mathematically independent and externally falsifiable via the performance illustrations on simulated data.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that reliable super-population forecasts exist and on standard credibility-theory variance components; no new entities are postulated and the only free parameters are the credibility weights derived from those variances.

free parameters (1)
  • credibility weights
    Weights in the weighted-average predictor are determined from variance estimates that are fitted to the data or model outputs.
axioms (1)
  • domain assumption Reliable forecasts for the super-population can be produced while sub-populations cannot be modelled reliably on their own.
    Explicitly stated as the foundational setup in the abstract.

pith-pipeline@v0.9.0 · 5759 in / 1235 out tokens · 113851 ms · 2026-05-19T04:06:48.905177+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
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    Relation between the paper passage and the cited Recognition theorem.

    sub-population credibility predictors that are weighted averages of expected future super-population mortality rates and expected future sub-population specific mortality rates... explicit expression for the expected quadratic forecast error... does not depend on the specific form of the super-population model

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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    A note on pandemic mortality rates

    Andersen, Per K, Borgan, Ornulf, Gill, Richard D, and Keiding, Niels (2012).Statistical models based on counting processes. Springer Science & Business Media. Andersson, Patrik and Lindholm, Mathias (2022). “A note on pandemic mortality rates”. In:Scan- dinavian Actuarial Journal2022.3, pp. 269–278. Booth, Heather, Maindonald, John, and Smith, Len (2002)....

    work page 2012

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    Evaluating and extending the Lee–Carter model for mortality forecasting: Bootstrap confidence interval

    John Wiley & Sons. Koissi, Marie-Claire, Shapiro, Arnold F, and Högnäs, Göran (2006). “Evaluating and extending the Lee–Carter model for mortality forecasting: Bootstrap confidence interval”. In:Insurance: Mathematics and Economics38.1, pp. 1–20. Lee, Ronald D and Carter, Lawrence R (1992). “Modeling and forecasting US mortality”. In:Journal of the Americ...

    work page doi:10.1093/jrsssa/ 2006

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    Model specification in multivariate time series

    Springer. Tiao, George C and Tsay, Ruey S (1989). “Model specification in multivariate time series”. In: Journal of the Royal Statistical Society: Series B (Methodological)51.2, pp. 157–195. Tsai, Cary Chi-Liang and Lin, Tzuling (2017). “Incorporating the Bühlmann credibility into mor- tality models to improve forecasting performances”. In:Scandinavian Ac...

    work page 1989

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    Further, we have that E h µx,t′+h Θi x −bθi xµx,t′+h µx,t′+h Θi x − µx,t′+h | G i x,t′ i = (σ2 x,t′+h + µ2 x,t′+h) Var Θi x + σ2 x,t′+h = Var µx,t′+h Θi x | G i x,t′

    + (µx,t′+h)2 Var Θi x + µx,t′+h Var Θi x P v(Ei x,vµx,v)2 (P v Eix,vµx,v)2 + 1P v Eix,vµx,v ! = Var Θi x P v(Ei x,vµx,v)2 (P v Eix,vµx,v)2 + σ2 x,t′+h + µx,t′+h ! + σ2 x,t′+h + 1P v Eix,vµx,v , with σ2 x,t′+h := Var(µx,t′+h | G i x,t′) = Var(µx,t′+h | µx,t, t ∈ T). Further, we have that E h µx,t′+h Θi x −bθi xµx,t′+h µx,t′+h Θi x − µx,t′+h | G i x,t′ i = ...

    work page 2021

  5. [5]

    In the picture they are represented represented in the square with vertices A,B,C,E

    The figure is an age-period representation of the tabular data for a periodt and age x. In the picture they are represented represented in the square with vertices A,B,C,E. However, the total individuals who have been exposed to the risk of death in periodt are those represented with the parallelogram A,C,D,E. Assuming a uniform distribution of the deaths...

    work page 2005