A Model-Agnostic Bootstrap for Macro-Level Claims Reserving Under the Conditioning Principle

Abstract

The correct inferential object in claims reserving is the conditional predictive distribution $p(R \mid \mathcal{D}, \hatθ)$, where $\mathcal{D}$ is the observed triangle held fixed. We refer to this as the conditioning principle. All existing bootstraps violate it by resampling functions of $\mathcal{D}$ inside the predictive loop, producing an $O(1)$ coverage error that does not vanish as the triangle grows. The Dirichlet-Gamma hierarchy admits a bootstrap that satisfies the principle exactly: $S^{IBNP}_i = X^{obs}_i (1-W_i)/W_i$ with $W_i \sim \mathrm{Beta}(c\hat{F}_{I-i}, c(1-\hat{F}_{I-i}))$ sampled directly from its predictive distribution. Only the allocation proportion $W_i$ is simulated; the observed triangle is held fixed. It thus inherits calibration from any development-proportion method (Chain-Ladder, Bornhuetter-Ferguson, Cape Cod, or other), making it model-agnostic. The coverage deficit is $O(I^{-1/2})$, independent of the number of development periods. Under compound Poisson data-generating processes the bootstrap is conservative for every $F_{I-i} \in (0,1)$: the predictive standard deviation analytically exceeds the true value by the factor $1/\sqrt{F_{I-i}}$. The ODP bootstrap violates the principle through two mechanisms in opposite directions: re-estimation inflates bootstrap variance under the ODP DGP, while missing accident-year frailty deflates it under frailty DGPs. The resulting coverage discrepancy is $Ω(1)$ regardless of $I$, providing a structural explanation for the cross-portfolio miscalibration heterogeneity documented by Meyers (2015). Chain-Ladder, Bornhuetter-Ferguson and Cape Cod emerge as credibility estimators under diffuse, informative and pooling priors respectively, with identical structure for counts and amounts. The concentration $c$ serves as a diagnostic: $\hat{c} < 30$ signals non-stationary development.