Severity estimation in dependent collective risk models

Technical Summary: Severity Estimation in Dependent Collective Risk Models

Problem Statement
The collective risk model (CRM) represents aggregate insurance losses as a random sum of individual claim severities. Standard estimation procedures assume independence between claim frequency ($N$) and severity ($X$), and independence among severities within a policy. Under these assumptions, the "margin-first" approach—fitting the severity distribution to the pooled claims across a portfolio—is consistent. However, real-world portfolios often exhibit dependence between frequency and severity, as well as dependence among severities within a policy.

When such dependence exists, the claims pooled across policies are no longer a sample from the marginal severity distribution ($F_X$). Instead, policies with higher claim counts contribute disproportionately more claims to the pool. This creates a size-biased sampling mechanism where the empirical distribution of pooled claims converges to the law of an arbitrary observed claim ($Y$), rather than the marginal severity law. Consequently, fitting a severity model directly to uncorrected pooled claims yields inconsistent estimates of the true severity parameters, leading to systematic bias that does not vanish with sample size.

Methodology
The paper proposes a composite likelihood estimation procedure to address this inconsistency. The methodology rests on two pillars:

1. Identification of the Data-Generating Mechanism: The authors prove that the empirical distribution of pooled claims converges almost surely to the cumulative distribution function (CDF) of an arbitrary observed claim, $F_Y$. This distribution is a size-biased mixture of the conditional severity distributions:
   $$F_Y(x) = \sum_{n=1}^{\infty} \frac{n P(N=n)}{E[N]} F_{X|N=n}(x)$$
   This result identifies the specific distribution that pooled claims actually sample, distinct from the marginal $F_X$.

2. Composite Likelihood Construction: Rather than attempting to maximize the full joint likelihood of the policy vectors (which is often intractable due to high-dimensional copula derivatives), the authors construct a composite likelihood based on two low-dimensional components:

   • The marginal likelihood of the claim counts ($N$).
   • The likelihood of the pooled claims based on the observed-claim density $f_Y$.

   The composite log-likelihood is the sum of the count log-likelihood and the log-likelihood of the pooled severities treated as independent draws from $f_Y$.

3. Asymptotic Theory and Variance Estimation: The estimator is shown to be consistent and asymptotically normal. Crucially, the asymptotic variance is derived using the Godambe information matrix (a "sandwich" estimator). The paper emphasizes that the sampling unit must be the policy, not the individual claim. Aggregating scores at the policy level is necessary to account for the dependence among claims within a policy and the random cluster size. Aggregating at the claim level underestimates variance and leads to undercoverage.

4. Stepwise Extension: For models requiring higher-order dependence parameters (e.g., within-policy severity dependence), the paper outlines a stepwise estimation procedure. This involves estimating frequency parameters first, then severity and frequency-severity dependence, and finally within-policy dependence using conditional margins. A stacked sandwich estimator propagates the uncertainty from earlier steps.

Key Contributions and Results

• Theoretical Proof of Inconsistency: The paper formally establishes that margin-first estimators (including Maximum Likelihood, IFM, and rank-based copula methods) applied to pooled claims converge to the Kullback–Leibler projection of the true severity distribution onto the fitted family, rather than the true parameters themselves.
• Closed-Form Solutions for Sarmanov CRM: The methodology is specialized to the Sarmanov collective risk model. Using a latent Bernoulli-mixture representation, the authors derive closed-form expressions for the observed-claim density $f_Y$, the conditional severity laws, and the aggregate mean. This tractability allows for efficient implementation of the composite likelihood.
• Admissibility Constraints: The paper provides explicit constraints on the dependence parameters (e.g., $\theta_{01}, \theta_{12}, \theta_{012}$) to ensure the resulting copula is valid, derived from the non-negativity of the latent Bernoulli probability mass function.
• Simulation Evidence:
  • Bias Correction: A simulation study demonstrates that naive pooled fitting produces significant bias in severity parameter estimates (e.g., Gamma shape and rate) that persists regardless of sample size. The proposed composite likelihood removes this bias.
  • Aggregate Mean: Interestingly, while the naive estimator fails to recover the true severity margin, it still estimates the aggregate mean ($E[S]$) consistently because $E[S] = E[N]E[Y]$, and the naive fit correctly captures the mean of the observed claims ($E[Y]$).
  • Inference Validity: The study confirms that policy-level Godambe intervals achieve nominal coverage (approx. 95%). In contrast, claim-level sandwich estimators (which ignore within-policy dependence) result in severe undercoverage (e.g., dropping to 84% in high-frequency scenarios).

Significance
The paper claims that its primary contribution is the identification of the size-biased nature of pooled claims in dependent CRMs and the provision of a consistent estimation framework that corrects for this bias. It argues that standard two-step procedures are fundamentally flawed in the presence of frequency-severity dependence. By utilizing the observed-claim law ($F_Y$) within a composite likelihood framework, the method recovers the true marginal severity parameters and dependence structures. The work bridges the gap between copula-based risk modeling and size-biased sampling theory, offering a practical solution for actuarial estimation where full joint likelihoods are intractable. The authors note that while the Sarmanov model is used for illustration, the size-bias argument applies generally to any copula family with computable conditional laws.