Tail persistence of extremes at finite and asymptotic levels with Bayesian hierarchical inference

Abstract

Tail persistence characterizes the relative escalation of observations beyond an extreme threshold and complements conventional summaries such as return levels and tail indices. Existing literature is mainly distribution-based and does not account for uncertainty, nonstationarity, or covariate effects. We develop a Bayesian framework for modeling tail persistence under generalized extreme value and generalized Pareto models. We first make explicit the connection between tail persistence and the extreme value index under regular variation, which identifies the asymptotic persistence with $1/(1-ξ)$ and links persistence to tail heaviness. We then derive closed-form expressions for persistence at finite quantile levels under both models and show that, for any location--scale extreme value family with a common shape parameter, finite-level persistence depends on the location and scale parameters only through their ratio. Motivated by these results and by the weak identifiability of observation-specific shape parameters, we formulate a Bayesian hierarchical extreme value model in which the shape parameter is constant over time, while the location and scale parameters vary with covariates and latent effects. The asymptotic persistence is then a single, well-identified quantity, whereas finite-level persistence evolves over time through the location--scale structure. The framework provides posterior inference for both quantities, with credible intervals and probabilistic assessment of temporal evolution and covariate effects. Through simulation studies and real-data applications, we show that persistence provides an interpretable characterization of the escalation of extremes beyond high thresholds, complementing classical extreme value summaries with comprehensive uncertainty quantification in nonstationary settings. The methodology is implemented in an open-source \texttt{R} package.