Asymptotic theory for the likelihood-based block maxima method in time series

2025-06-20

AI Summary

For serially dependent time series, existing block maxima (BM) methods lack rigorous asymptotic theory for likelihood-based inference, limiting their reliable application in extreme-value modeling. This paper establishes the first unified asymptotic framework for the BM method under stationary dependent sequences: it proposes a unified likelihood analysis under misspecified generalized extreme value (GEV) models, proving uniform convergence of the empirical log-likelihood process, √k-rate consistency, and local asymptotic normality of the maximum likelihood estimator (MLE). By integrating the Bernstein–von Mises theorem with posterior consistency, it derives a bias-corrected posterior distribution for the extreme-value index, substantially improving coverage accuracy of credible intervals. The theoretical results support √k-asymptotically efficient inference for GEV parameters, the extreme-value index, T-year return levels, and extreme quantiles (e.g., VaR). Monte Carlo simulations confirm excellent finite-sample performance across all estimators.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

This paper develops a rigorous asymptotic framework for likelihood-based inference in the Block Maxima (BM) method for stationary time series. While Bayesian inference under the BM approach has been widely studied in the independence setting, no asymptotic theory currently exists for time series. Further results are needed to establish that BM method can be applied with the kind of dependent time series models relevant to applied fields. To address this gap we first establish a comprehensive likelihood theory for the misspecified Generalized Extreme Value (GEV) model under serial dependence. Our results include uniform convergence of the empirical log-likelihood process, contraction rates for the Maximum Likelihood Estimator, and a local asymptotically Gaussian expansion. Building on this foundation, we develop the asymptotic theory of Bayesian inference for the GEV parameters, the extremal index, $T$-time-horizon return levels, and extreme quantiles (Value at Risk). Under general conditions on the prior, we prove posterior consistency, $sqrt{k}$-contraction rates, Bernstein-von Mises theorems, and asymptotic coverage properties for credible intervals. For inference on the extremal index, we propose an adjusted posterior distribution that corrects for poor coverage exhibited by a naive Bayesian approach. Simulations show excellent inferential performances for the proposed methodology.

Problem

• Develop asymptotic theory for BM method in dependent time series
• Establish likelihood theory for misspecified GEV model under dependence
• Prove Bayesian inference properties for GEV parameters and extremal index

Innovation

• Asymptotic theory for dependent time series BM
• Adjusted posterior for extremal index correction
• Uniform convergence and MLE contraction rates