Submitted on 2019-10-16

Bias resulting from model misspecification is a concern when predicting insurance claims. Indeed, this bias puts the insurer at risk of making invalid or unreliable predictions. A method that could provide provably valid predictions uniformly across a large class of possible distributions would effectively eliminate the risk of model misspecification bias. Conformal prediction is one such method that can meet this need, and here we tailor that approach to the typical insurance application and show that the predictions are not only valid but also efficient across a wide range of settings.

Submitted on 2019-07-19

In the context of predicting future claims, a fully Bayesian analysis---one that specifies a statistical model, prior distribution, and updates using Bayes's formula---is often viewed as the gold-standard, while Buhlmann's credibility estimator serves as a simple approximation. But those desirable properties that give the Bayesian solution its elevated status depend critically on the posited model being correctly specified. Here we investigate the asymptotic behavior of Bayesian posterior distributions under a misspecified model, and our conclusion is that misspecification bias generally has damaging effects that can lead to inaccurate inference and prediction. The credibility estimator, on the other hand, is not sensitive at all to model misspecification, giving it an advantage over the Bayesian solution in those practically relevant cases where the model is uncertain. This begs the question: does robustness to model misspecification require that we abandon uncertainty quantification based on a posterior distribution? Our answer to this question is *No*, and we offer an alternative *Gibbs posterior* construction. Furthermore, we argue that this Gibbs perspective provides a new characterization of Buhlmann's credibility estimator.

This note generalizes the notion of conditional probability to Riesz spaces using the order-theoretic approach. With the aid of this concept, we establish the law of total probability and Bayes' theorem in Riesz spaces; we also prove an inclusion-exclusion formula in Riesz spaces. Several examples are provided to show that the law of total probability, Bayes' theorem and inclusion-exclusion formula in probability theory are special cases of our results.

Submitted on 2018-09-04

Accurate estimation of value-at-risk (VaR) and assessment of associated uncertainty is crucial for both insurers and regulators, particularly in Europe. Existing approaches link data and VaR indirectly by first linking data to the parameter of a probability model, and then expressing VaR as a function of that parameter. This indirect approach exposes the insurer to model misspecification bias or estimation inefficiency, depending on whether the parameter is finite- or infinite-dimensional. In this paper, we link data and VaR directly via what we call a discrepancy function, and this leads naturally to a Gibbs posterior distribution for VaR that does not suffer from the aforementioned biases and inefficiencies. Asymptotic consistency and root-*n* concentration rate of the Gibbs posterior are established, and simulations highlight its superior finite-sample performance compared to other approaches.

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