Bayesian Estimation for a Poisson Claim Count with Heterogeneity

Summary

The document sets up a hierarchical claim-count model for non-life insurance. Conditional on a positive latent heterogeneity parameter, the number of claims is Poisson distributed; the parameter itself has a continuous density on the positive real line. The author notes the conditional mean and asks for the posterior density of the latent parameter after observing a claim count.

The problem also proposes identities connecting the posterior mean to adjacent marginal claim-count probabilities, and higher conditional moments to a product of shifted posterior means. These are posed as results to derive or verify, not established in the document. No prior distribution, solution, proof, numerical example, or discussion of when the identities apply is provided. The setup is relevant to Bayesian frequency modeling and credibility-style estimation, but the text alone does not supply a complete method or validate the proposed moment formula.

Key ideas

• The claim count is modeled as Poisson conditional on a latent positive heterogeneity parameter.
• The parameter has a continuous prior density, and the target is its posterior distribution given the observed count.
• The document proposes a relationship between the posterior mean and adjacent marginal probabilities of the claim count.
• A product identity for higher posterior moments is posed for verification but is not proved in the document.

Tags

# Poisson distributed claim in non life insurance mathematics


# Poisson distributed claim in non life insurance mathematics












I am struggling with the following problem. I assume that there is a single claim number $X$ with corresponding heterogeneity parameter $\theta>0$.

I assume that $X$ given $\theta$ is Poisson distributed with parameter $\theta$, where $\theta$ has a continuous density $f_{\theta}$ on $(0,\infty)$ and I have derived that $\mathbb{E}[X|\theta]=\theta$.

I want to determine the conditional density $f_{\theta}(y|X=k)$ of $\theta$ given $X$ and I think that can be applied to calculate the Bayes estimator $m_k=\mathbb{E}[\theta|X=k]$.

Furthermore, I want to show that it holds that,

$$ m_k=(k+1)\frac{P(X=k+1)}{P(X=k)} $$

and verify that for $n\geq 1m$ it holds that,

$$ \mathbb{E}[\theta^n|X=k]=\prod_{j=1}^{n-1}m_{k+j} $$

Thanks in advance-

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