A hybrid-Hill estimator enabled by heavy-tailed block maxima
Abstract
When analysing extreme values, two alternative statistical approaches have historically been held in contention: the seminal block maxima method (or annual maxima method, spurred by hydrological applications) and the peaks-over-threshold. Clamoured amongst statisticians as wasteful of potentially informative data, the block maxima method gradually fell into disfavour whilst peaks-over-threshold-based methodologies were ushered to the centre stage of extreme value statistics. This paper proposes a hybrid method which reconciles these two hitherto disconnected approaches. Appealing in its simplicity, our main result introduces a new universal limiting characterisation of extremes that eschews the customary requirement of a sufficiently large block size for the plausible block maxima-fit to an extreme value distribution. We advocate that inference should be drawn solely on larger block maxima, from which practice the mainstream peaks-over-threshold methodology coalesces. The asymptotic properties of the promised hybrid-Hill estimator herald more than its efficiency, but rather that a fully-fledged unified semi-parametric stream of statistics for extreme values is viable. A finite sample simulation study demonstrates that a reduced-bias off-shoot of the hybrid-Hill estimator fares exceptionally well against the incumbent maximum likelihood estimation that relies on a numerical fit to the entire sample of block maxima.
Summary
This paper addresses the problem of statistically analyzing extreme values, specifically reconciling two historically disconnected approaches: the block maxima (BM) method and the peaks-over-threshold (POT) method. The authors propose a hybrid method that combines these approaches, introducing a new limiting characterization of extremes that doesn't require large block sizes for extreme value distribution fitting. This method advocates for inference based on larger block maxima, which aligns with the mainstream POT methodology. The core of their approach is a new hybrid-Hill estimator. The authors demonstrate through simulations that a reduced-bias version of their estimator performs well against maximum likelihood estimation, which relies on fitting an extreme value distribution to the entire sample of block maxima. This research matters because it offers a unified semi-parametric framework for extreme value statistics, potentially improving the accuracy and efficiency of extreme value analysis. The paper's methodology involves a theoretical development of the hybrid-Hill estimator and its asymptotic properties. It starts with the classical BM method and builds a bridge to the POT method by focusing on the tail distribution function of block maxima. Conditions A1 and B are introduced to characterize the max-domains of attraction, unifying BM and POT conditions. The asymptotic normality of the estimator is proven under certain conditions, and the paper explores optimal choices for the top fraction of block maxima used in the estimation. Finally, a bias reduction procedure is devised using a cursory estimator to improve the performance of the hybrid-Hill estimator. The paper provides a comprehensive simulation study to validate the theoretical findings.
Key Insights
- •Hybrid-Hill Estimator: The paper introduces a novel hybrid-Hill estimator (H2) that merges block maxima (BM) and peaks-over-threshold (POT) methodologies for extreme value analysis.
- •Universal Limiting Characterization: The authors establish a new universal limiting characterization of extremes, removing the requirement of sufficiently large block sizes for extreme value distribution fitting.
- •Asymptotic Properties: The paper derives the asymptotic properties of the hybrid-Hill estimator, demonstrating its efficiency and potential for a unified semi-parametric approach to extreme values.
- •Bias Reduction: A reduced-bias off-shoot of the hybrid-Hill estimator is developed, which demonstrably outperforms incumbent maximum likelihood estimation, especially with smaller block sizes.
- •Universality Class: Lemma 5 establishes a distribution-free result amenable to the universality class of distributions induced by the new extreme value condition (1.6).
- •Optimal Choice of k0: The paper provides guidance on the optimal selection of the top fraction of block maxima (k0/k) to minimize the asymptotic mean squared error (AMSE) of the H2 estimator, considering cases with different rates of convergence (tilde{rho} = -1 and -1 < tilde{rho} <= 0).
- •Second Order Refinement: Condition A2 is introduced to capture the approximation bias, a deterministic pre-asymptotic bias, that deviations from the limiting Pareto-tail instil in the H2 estimator.
Practical Implications
- •Improved Extreme Value Analysis: Practitioners in fields such as hydrology, finance, and climate science can use the hybrid-Hill estimator to improve the analysis of extreme events.
- •Unified Framework: The unified semi-parametric framework simplifies extreme value analysis by combining two previously separate methodologies.
- •Reduced Bias Estimation: The reduced-bias estimator offers more accurate results, particularly when data is limited or block sizes are small.
- •Data-Driven Simplicity: The hybrid-Hill estimator is amenable to a wide range of applications due to its generality and data-driven simplicity.
- •Future Research: The work opens avenues for further research in several directions, including the development of statistical methods for trends in extremes and the application of the hybrid framework to non-stationary block maxima.