An inferential model encodes the data analyst's degrees of belief about an unknown quantity of interest based on the observed data, posited statistical model, etc. Inferences drawn based on these degrees of belief should be reliable in a certain sense, so we require the inferential model to be valid. The construction of valid inferential models based on individual pieces of data is relatively straightforward, but how to combine these so that the validity property is preserved? In this paper we analyze some common combination rules with respect to this question, and we conclude that the best strategy currently available is one that combines via a certain dimension reduction step before the inferential model construction.
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.