<p>The extremal index is an important characteristic measuring dependence of extreme values in a stationary series. Several new estimators that are mostly based on interexceedance times within the Peaks-over-Threshold model have been recently published. Nevertheless, in many cases these estimators rely on suitable choice of auxiliary parameters and/or are derived under assumptions that are related to validity of the local dependence condition <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10687_2025_513_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$D^{(k)}(u_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>D</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Although the determination of the correct order <i>k</i> in the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10687_2025_513_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$D^{(k)}(u_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>D</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> condition can have major effect on the extremal index estimates, there are not many reliable methods available for this task. In this paper, we present various approaches to assessing validity of the <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10687_2025_513_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$D^{(k)}(u_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>D</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> condition including a graphical diagnostics and propose several statistical tests. A simulation study is carried out to determine performance of the statistical tests, particularly the type I and type II errors.</p>

Statistical inference on the local dependence condition of extreme values in a stationary sequence

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2025年5月23日

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Jan Holešovský,
Michal Fusek

摘要

The extremal index is an important characteristic measuring dependence of extreme values in a stationary series. Several new estimators that are mostly based on interexceedance times within the Peaks-over-Threshold model have been recently published. Nevertheless, in many cases these estimators rely on suitable choice of auxiliary parameters and/or are derived under assumptions that are related to validity of the local dependence condition $D^{(k)}(u_n)$ $D^{(k)} (u_{n})$ . Although the determination of the correct order k in the $D^{(k)}(u_n)$ $D^{(k)} (u_{n})$ condition can have major effect on the extremal index estimates, there are not many reliable methods available for this task. In this paper, we present various approaches to assessing validity of the $D^{(k)}(u_n)$ $D^{(k)} (u_{n})$ condition including a graphical diagnostics and propose several statistical tests. A simulation study is carried out to determine performance of the statistical tests, particularly the type I and type II errors.