A modified generalized pareto distribution with a three-level hierarchical bayesian estimation and financial application

A comprehensive framework for extreme value analysis grounded in hierarchical Bayesian modeling of the Generalized Pareto Distribution (HB-GPD). This study proposes a robust and computationally intensive framework for extreme value analysis via a three-tier HB-GPD. The model rigorously captures within-group volatility and cross-group heterogeneity through structured prior and hyperprior hierarchies. To address the limitations of classical estimators under sparse data and heavy-tailed regimes, we compare Maximum Likelihood Estimation (MLE), Method of Moments (MoM), Probability-Weighted Moments (PWM), and Empirical Percentile Method (EPM) against our Bayesian paradigm. Posterior inference is conducted using advanced Markov Chain Monte Carlo (MCMC) techniques, including Metropolis-Hastings within Gibbs sampling and the No-U-Turn Sampler (NUTS), ensuring efficient posterior exploration in high-dimensional spaces. Asymptotic Relative Efficiency (ARE) as performance diagnostics. Simulation studies and empirical financial data from Nifty 50 and S&P 500 sectors substantiate the model's superiority in estimating Value-at-Risk and Expected Shortfall, thereby affirming its relevance in actuarial science, operational risk, and financial solvency analytics.