Severity estimation in dependent collective risk models

The collective risk model represents the aggregate loss of an insurance portfolio as a random sum of individual claim severities. When claim counts and severities are dependent, the claims pooled across policies are no longer a sample from the marginal severity distribution. We show that their empirical distribution converges to the law of an arbitrary observed claim, a size-biased mixture of the conditional severity distributions, so any procedure that fits the severity margin directly to pooled claims is inconsistent in general. The same result identifies the distribution that the pooled claims do sample, and we build a composite likelihood estimation procedure on that distribution. We establish consistency and asymptotic normality, with Godambe information in which the policy, rather than the claim, is the sampling unit. In a Sarmanov collective risk model, the observed-claim density and the aggregate mean are in closed form. A simulation study measures the bias of naive pooled-severity fitting, its correction by the composite likelihood, and the coverage of the policy-level standard errors.