<p>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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ell ^p-$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mi>p</mi> </msup> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>norms, for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. The case <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p=\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is the tail index of the time series, has specific nice properties thus we analyze the asymptotics of block estimators when approximating <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> 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 (Probab. Theory Related Fields <b>95</b>, 331–356 <CitationRef CitationID="CR26">1993</CitationRef>). We illustrate our findings on simulations.</p>

On the asymptotics of extremal $\ell ^p-$blocks cluster inference

期刊论文
出版模式：

开放获取
发表日期：

2025年6月28日

查看全文

Gloria Buriticá,
Olivier Wintenberger

摘要

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 $\ell ^p-$ $ℓ^{p} -$ norms, for $p > 0$ $p > 0$ . The case $p=\alpha $ $p = α$ , where $\alpha > 0$ $α > 0$ is the tail index of the time series, has specific nice properties thus we analyze the asymptotics of block estimators when approximating ${\alpha }$ $α$ 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 (Probab. Theory Related Fields 95, 331–356 1993). We illustrate our findings on simulations.

On the asymptotics of extremal \(\ell ^p-\)blocks cluster inference

摘要