PickandsBalkemade

PickandsBalkemade is a hypothetical statistical framework in extreme value theory designed to model rare, high-impact events across univariate and multivariate data streams. It extends the threshold exceedance paradigm by coupling a flexible tail model with covariate-driven structure to capture nonstationary tail behavior. The framework aims to provide robust estimates of tail risk, such as high quantiles and exceedance probabilities, with interpretable parameters.

The theoretical core generalizes the peaks-over-threshold approach by allowing the exceedance distribution above a high threshold to follow a parametric tail family whose shape and scale depend on covariates. It introduces a tail dependence function for multivariate extremes and specifies identifiability and regularity conditions. The model accommodates both stationary and nonstationary settings and supports modular extensions for different data-generating processes.

Estimation commonly uses local likelihood or Bayesian methods, with data-driven threshold selection and diagnostic checks. Software implementations are described for R and Python, providing tools for parameter estimation, model comparison, and out-of-sample validation. Applications span finance risk management, hydrology, environmental science, and climate risk assessment.

Limitations and related topics. As with other threshold-based approaches, performance is sensitive to threshold choice and data sparsity in the tail. The framework relates to established extreme value theory, including the Pickands–Balkema–de Haan theorem and generalized Pareto modeling, as well as multivariate tail dependence modeling. See also extreme value theory, generalized Pareto distribution, threshold selection.