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On the asymptotics of extremal lp-blocks cluster inference

Gloria Buriticá, Olivier Wintenberger

TL;DR

This work develops a comprehensive framework for inferring extremal clustering in stationary regularly varying time series through $\ell^p$-block statistics. By establishing asymptotic normality for disjoint-block estimators based on extreme block observations and by plugging in Hill estimates for the tail index, the authors unify inference for a broad class of cluster functionals, including the extremal index, cluster-size probabilities, and cluster indices for sums. They verify the theory on linear models and stochastic recurrence equations under Kesten’s conditions, showing potential variance reduction relative to traditional block methods, and support the results with numerical experiments. The approach provides practical, variance-aware tools for extremal clustering analysis in heavy-tailed time series with robust tuning guidance for finite samples.

Abstract

Extremes occur in stationary regularly varying time series as short periods with several large observations, known as extremal blocks. We study cluster statistics summarizing the behavior of functions acting on these extremal blocks. Examples of cluster statistics are the extremal index, cluster size probabilities, and other cluster indices. The purpose of our work is twofold. First, we state the asymptotic normality of block estimators for cluster inference based on consecutive observations with large lp-norms, for p < 0. The case p=$α$, where $α$ > 0 is the tail index of the time series, has specific nice properties thus we analyze the asymptotic of blocks estimators when approximating $α$ using the Hill estimator. Second, we verify the conditions we require on classical models such as linear models and solutions of stochastic recurrence equations. Regarding linear models, we prove that the asymptotic variance of classical index cluster-based estimators is null as first conjectured in Hsing T. [26]. We illustrate our findings on simulations.

On the asymptotics of extremal lp-blocks cluster inference

TL;DR

This work develops a comprehensive framework for inferring extremal clustering in stationary regularly varying time series through -block statistics. By establishing asymptotic normality for disjoint-block estimators based on extreme block observations and by plugging in Hill estimates for the tail index, the authors unify inference for a broad class of cluster functionals, including the extremal index, cluster-size probabilities, and cluster indices for sums. They verify the theory on linear models and stochastic recurrence equations under Kesten’s conditions, showing potential variance reduction relative to traditional block methods, and support the results with numerical experiments. The approach provides practical, variance-aware tools for extremal clustering analysis in heavy-tailed time series with robust tuning guidance for finite samples.

Abstract

Extremes occur in stationary regularly varying time series as short periods with several large observations, known as extremal blocks. We study cluster statistics summarizing the behavior of functions acting on these extremal blocks. Examples of cluster statistics are the extremal index, cluster size probabilities, and other cluster indices. The purpose of our work is twofold. First, we state the asymptotic normality of block estimators for cluster inference based on consecutive observations with large lp-norms, for p < 0. The case p=, where > 0 is the tail index of the time series, has specific nice properties thus we analyze the asymptotic of blocks estimators when approximating using the Hill estimator. Second, we verify the conditions we require on classical models such as linear models and solutions of stochastic recurrence equations. Regarding linear models, we prove that the asymptotic variance of classical index cluster-based estimators is null as first conjectured in Hsing T. [26]. We illustrate our findings on simulations.
Paper Structure (57 sections, 23 theorems, 282 equations, 4 figures)

This paper contains 57 sections, 23 theorems, 282 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Mixing coefficients
  5. Regular variation
  6. $\ell^p-$cluster processes
  7. Asymptotics of $p-$cluster block estimators
  8. Block estimators
  9. Tail-index estimation
  10. Consistency of block estimators
  11. Assumptions for asymptotic normality
  12. Asymptotic normality of block estimators
  13. Cluster statistics
  14. The extremal index
  15. The cluster index for sums
  16. ...and 42 more sections

Key Result

Proposition 2.2

Let $(\mathbf{X}_t)$ be a stationary time series satisfying $\bf RV_\alpha$. Let $(x_n)$ be a sequence such that $n\,{\mathbb{P}}(|\mathbf{X}_1|> x_n) \to 0$, as $n \to \infty$, and $p > 0$. For all $\epsilon > 0$, $\delta > 0$, assume $\lim_{s\to \infty} \limsup_{n \to \infty} {\mathbb{P}}(\|\mathb

Figures (4)

  • Figure 1: Heatmap with contour curves of standard deviations and mean squared errors for estimates of the extremal index $k1 = k \mapsto \widehat{\theta}_{X}(k)$ in \ref{['eq:estimator:Ei']}, using the Hill estimator $k2 = k^\prime \mapsto \widehat{\alpha}(k^\prime)$. We simulate $500$ samples $(X_t)_{t=1,\dots,n}$ of an AR$(\varphi)$ model with absolute value student$(\alpha)$ noise for $n = 12\,000$, $\varphi = 0.5$, $\alpha = 1$, such that $\theta_{X} = 0.5$.
  • Figure 2: Heatmap with contour curves as in Figure \ref{['fig:0.78000']}. Here we simulate $500$ samples $(X_t)_{t=1,\dots,n}$ of an AR$(\varphi)$ model with absolute value student$(\alpha)$ noise for $n = 12\,000$, $\varphi = 0.7$, $\alpha = 1$, such that $\theta_{X} = 0.3$.
  • Figure 3: Heatmap with contour curves as in Figure \ref{['fig:0.78000']}. Here we simulate $500$ samples $(X_t)_{t=1,\dots,n}$ of Example \ref{['ex:sre']} for $n = 12\,000$ such that $\theta_{X} \approx 0.2792$.
  • Figure 4: Histogram of estimates $\widehat{\theta}_X$ of the extremal index using \ref{['eq:estimator:Ei']}, and the cluster size probability $\widehat{\pi}_1$, $\widehat{\pi}_2$, $\widehat{\pi}_3$, using \ref{['eq:xmas25c']}. We simulate $1\,000$ samples $(X_t)_{t=1,\dots,n}$ of Example \ref{['ex:sre']} with $n = 12\,000$. The Gaussian density curves are centered in the median of the estimators. Their variances are estimated by Monte-Carlo (dotted curve) or using the average of the cluster-based estimate of the asymptotic variance defined in \ref{['eq:asymptotic:variance:pi']} (solid curve). The red lines point to the Monte-Carlo approximation of the real values with standard deviation. These were computed using Equation 3.5 in haan:resnick:rootzen:vries:1989, and a simulation study with $10\,000$ samples of length $500\,000$.

Theorems & Definitions (39)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Lemma 3.1
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • Corollary 4.1
  • proof
  • Remark 4.2
  • ...and 29 more