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A hybrid-Hill estimator enabled by heavy-tailed block maxima

Claudia Neves, Chang Xu

TL;DR

This work addresses the historical split between block maxima and peaks-over-threshold approaches to extremes by introducing a hybrid-Hill estimator (H2) that uses heavy-tailed block maxima within a semi-parametric max-domain framework. The method relies on Conditions $A1$, $A2$, and $B$ and leverages the tail behavior of $F^m$ to unify BM and POT tails, yielding a consistent estimator for the positive extreme value index $\gamma>0$ and an asymptotically normal distribution for $\sqrt{k_0}(\hat{\gamma}^{\mathrm{H2}}-\gamma)$. It also develops an optimal fraction $k_0/k$ selection and a reduced-bias variant (RBH2) that often outperforms the traditional MLE and conventional Hill-type estimators, particularly in finite samples and under dependence/non-stationarity. The approach offers a practical, robust, semi-parametric framework for extreme value inference with broad applicability and potential extensions to dependent and non-stationary data settings.

Abstract

When analysing extreme values, two alternative statistical approaches have historically been held in contention: the 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 climbed to the centre stage of extreme value statistics. This paper devises a hybrid method which reconciles these two hitherto disconnected approaches. Appealing in its simplicity, our main result introduces a new universality class of extreme value distributions that discards the customary requirement of a sufficiently large block size for the plausible block maxima-fit to an extreme value distribution. Natural extensions to dependent and/or non-stationary settings are mapped out. 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 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 reduced-bias off-shoot of the hybrid-Hill estimator provably outclasses the incumbent maximum likelihood estimation that relies on a numerical fit to the entire sample of block maxima.

A hybrid-Hill estimator enabled by heavy-tailed block maxima

TL;DR

This work addresses the historical split between block maxima and peaks-over-threshold approaches to extremes by introducing a hybrid-Hill estimator (H2) that uses heavy-tailed block maxima within a semi-parametric max-domain framework. The method relies on Conditions , , and and leverages the tail behavior of to unify BM and POT tails, yielding a consistent estimator for the positive extreme value index and an asymptotically normal distribution for . It also develops an optimal fraction selection and a reduced-bias variant (RBH2) that often outperforms the traditional MLE and conventional Hill-type estimators, particularly in finite samples and under dependence/non-stationarity. The approach offers a practical, robust, semi-parametric framework for extreme value inference with broad applicability and potential extensions to dependent and non-stationary data settings.

Abstract

When analysing extreme values, two alternative statistical approaches have historically been held in contention: the 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 climbed to the centre stage of extreme value statistics. This paper devises a hybrid method which reconciles these two hitherto disconnected approaches. Appealing in its simplicity, our main result introduces a new universality class of extreme value distributions that discards the customary requirement of a sufficiently large block size for the plausible block maxima-fit to an extreme value distribution. Natural extensions to dependent and/or non-stationary settings are mapped out. 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 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 reduced-bias off-shoot of the hybrid-Hill estimator provably outclasses the incumbent maximum likelihood estimation that relies on a numerical fit to the entire sample of block maxima.
Paper Structure (14 sections, 10 theorems, 94 equations, 5 figures)

This paper contains 14 sections, 10 theorems, 94 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Notation
  4. N.N.I.I.D. random variables or I.N.N.I.D. random variables
  5. A cursory semi-parametric hybrid estimator
  6. Hybrid-Hill (H2) estimator based on $k_0$ tail-related BM
  7. Consistency
  8. Asymptotic normality
  9. Optimal choices for the top fraction $k_0/k$ in H2 estimation
  10. Reduced-bias H2 estimation
  11. Proofs
  12. Results concerning Condition B
  13. Second order regular variation development of Condition A2
  14. First and second order regular variation of $1-F^m$

Key Result

Theorem 1

Let $\{X_n\}_{n\geq 1}$ be a sequence of i.i.d. random variables drawn from a distribution $F$ such that Condition A1 holds. For any block length $m\geq 2$, the cursory estimator $\hat{\gamma}(\theta_k, k)$ defined in eq:EstNaive is a consistent estimator for $\gamma >0$ in the sense that, as $k \ri provided $\theta_k \in (0, k]$ is an upper intermediate sequence, i.e., such that $\theta_k \righta

Figures (5)

  • Figure 1: Average estimates of both hybrid-Hill and GEV-maximum likelihood estimates and their respective empirical mean squared errors (MSE), plotted against the block size $m=1, \ldots, 100$. Simulations consist of $1000$ replicates of a sample of $n=10,000$ i.i.d. observations taken from a Pareto distribution with extreme value index $\gamma =1/4$.
  • Figure 2: Fréchet and Pareto parent distributions, both attached to $\gamma=0.3$ and $\tilde{\rho}= -1$. Inference produced by H2 estimators is virtually the same for these two distributions.
  • Figure 3: Burr distribution satisfying \ref{['2RVlogVg']} with $\gamma= 1/(\tau \lambda)= 0.75,\, 0.3$ and $\tilde{\rho} = -1/\lambda =-1,\, -2$, respectively top and bottom rows. Reduced-bias H2 (RBH2r) estimation is demonstrated accurate and surpasses the MLE for a wide range of intermediate values $k_0$. This is in spite of the MLE's smaller asymptotic variance for $\gamma \geq 1$ (top) and the misspecification of $\tilde{\rho}=-1$ to shorten H2 of its asymptotic bias (bottom).
  • Figure 4: Inference produced by H2 estimation based on the limiting GEV ascribed to BM and the stable GPD for POT exceedances, both of which here exemplified with $\gamma=0.25$ and $\tilde{\rho} = -\gamma$, is virtually the same.
  • Figure 5: Purely POT-GPD: conventional Hill estimation (i.e. H2 with $m=1$) for threshold exceedances on the basis of samples of size $n=5000$ from the GDP with $\gamma=0.25$ and $\tilde{\rho}= -\gamma$.

Theorems & Definitions (19)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • Theorem 4
  • Lemma 5
  • Remark 1
  • Remark 2
  • Theorem 6
  • Remark 3
  • ...and 9 more