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Counting models with excessive zeros ensuring stochastic monotonicity

Hyemin Lee, Dohee Kim, Banghee So, Jae Youn Ahn

TL;DR

This work investigates whether counting models for excessive zeros used in insurance pricing preserve stochastic monotonicity, a principle that posterior risk should increase with past claims. It first shows that Poisson–hurdle mixtures with correlated random effects can violate the credibility order, undermining fair posteriors. It then develops two monotonicity-guaranteeing classes: independent random effects (with Beta–Bernoulli and Gamma–Poisson conjugacy) offer base-order guarantees, and comonotonic random effects ensure both base and general credibility orders under suitable conditions. An empirical study on the LGPIF dataset demonstrates that the comonotonic model achieves the best predictive performance while preserving monotonicity, suggesting a practically robust approach for credibility adjustments in claim-frequency modeling. The results extend to NB and zero-inflated variants, indicating a broadly applicable framework for theoretically coherent zero-heavy count models in actuarial practice.

Abstract

Standard count models such as the Poisson and Negative Binomial models often fail to capture the large proportion of zero claims commonly observed in insurance data. To address such issue of excessive zeros, zero-inflated and hurdle models introduce additional parameters that explicitly account for excess zeros, thereby improving the joint representation of zero and positive claim outcomes. These models have further been extended with random effects to accommodate longitudinal dependence and unobserved heterogeneity. However, their consistency with fundamental probabilistic principles in insurance, particularly stochastic monotonicity, has not been formally examined. This paper provides a rigorous analysis showing that standard counting random-effect models for excessive zeros may violate this property, leading to inconsistencies in posterior credibility. We then propose new classes of counting random-effect models that both accommodate excessive zeros and ensure stochastic monotonicity, thereby providing fair and theoretically coherent credibility adjustments as claim histories evolve.

Counting models with excessive zeros ensuring stochastic monotonicity

TL;DR

This work investigates whether counting models for excessive zeros used in insurance pricing preserve stochastic monotonicity, a principle that posterior risk should increase with past claims. It first shows that Poisson–hurdle mixtures with correlated random effects can violate the credibility order, undermining fair posteriors. It then develops two monotonicity-guaranteeing classes: independent random effects (with Beta–Bernoulli and Gamma–Poisson conjugacy) offer base-order guarantees, and comonotonic random effects ensure both base and general credibility orders under suitable conditions. An empirical study on the LGPIF dataset demonstrates that the comonotonic model achieves the best predictive performance while preserving monotonicity, suggesting a practically robust approach for credibility adjustments in claim-frequency modeling. The results extend to NB and zero-inflated variants, indicating a broadly applicable framework for theoretically coherent zero-heavy count models in actuarial practice.

Abstract

Standard count models such as the Poisson and Negative Binomial models often fail to capture the large proportion of zero claims commonly observed in insurance data. To address such issue of excessive zeros, zero-inflated and hurdle models introduce additional parameters that explicitly account for excess zeros, thereby improving the joint representation of zero and positive claim outcomes. These models have further been extended with random effects to accommodate longitudinal dependence and unobserved heterogeneity. However, their consistency with fundamental probabilistic principles in insurance, particularly stochastic monotonicity, has not been formally examined. This paper provides a rigorous analysis showing that standard counting random-effect models for excessive zeros may violate this property, leading to inconsistencies in posterior credibility. We then propose new classes of counting random-effect models that both accommodate excessive zeros and ensure stochastic monotonicity, thereby providing fair and theoretically coherent credibility adjustments as claim histories evolve.
Paper Structure (21 sections, 11 theorems, 100 equations, 10 tables)

This paper contains 21 sections, 11 theorems, 100 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Violation of credibility order in Poisson-hurdle mixture model
  4. Limitations of the Poisson--hurdle mixture model with correlated random effects
  5. Numerical examples
  6. Poisson–hurdle mixture model with the independent random effects
  7. Poisson–hurdle mixture model with comonotonic random effects
  8. Empirical study: LGPIF insurance data
  9. Model setting
  10. Numerical investigation of the credibility order
  11. Out-of-sample validation
  12. Conclusion
  13. Proofs
  14. Proof of Lemma \ref{['lem.2']}
  15. Proof of Theorem \ref{['thm.1']}
  16. ...and 6 more sections

Key Result

Lemma 1

Let $\Theta$ be a latent random variable with cumulative distribution function $G$ supported on $\mathcal{T}\subseteq\mathbb R$. Conditional on $\Theta=\theta$, let $X_1,\dots,X_t$ be independent random variables with conditional distribution functions $F_j(\cdot\mid\theta)$ and corresponding densit If, for each $j$, $f_j(x_j\mid\theta)$ is TP$_2$ in $(x_j,\theta)$, then $X_{1:t}$ is MTP$_2$.

Theorems & Definitions (24)

  • Lemma 1: denuit2006actuarial
  • Lemma 2
  • proof
  • Lemma 3
  • Theorem 1
  • Remark 1
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • ...and 14 more