A collective risk model with shared latent variable inducing dependence among claim frequency and sizes / A. Barbiero, A. Hitaj. 38. Panhellenic and 4th International Statistics Conference Salonicco 2026.
A collective risk model with shared latent variable inducing dependence among claim frequency and sizes
A. Barbiero
Primo
;
2026
Abstract
We propose a modification of the classical collective risk model that allows for dependence between claim frequency and claim sizes through a shared latent continuous random variable. Specifically, the claim number N and the individual claim amounts Xi are conditionally independent given a latent variable Z, assumed to follow a gamma distribution. Conditional on Z = z, the claim count N follows a Poisson distribution with parameter z, while the claim sizes Xi are independent and identically distributed according to an exponential distribution with rate parameter z. For the proposed model, we derive the distribution of the aggregate claim amount, analyze the dependence structure induced among claim sizes, and obtain the conditional distribution of claim amounts given the number of claims. We also derive an expression for the linear correlation between claim frequency and claim severity, which is, by construction, always negative. An application to a real insurance dataset illustrates the practical relevance of the model and confirms its ability to capture frequency–severity dependence.
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