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Abstract <p>Grubbs’s test statistics are studied, that is, absolute values of extreme studentized deviations of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\)</EquationSource> <!--RusMath2570072Shiryaeva-m1--> </InlineEquation> random observations from the mean. We consider the case when random observations have arbitrary continuous marginal distributions. The existence of two regions is proved; in one of them, the joint distribution function of these statistics is a linear function of their marginal distribution functions, and in the other, the joint distribution function is zero. We construct a Grubbs’s copula from the joint distribution of Grubbs’s statistics. For the case <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n &gt; 3\)</EquationSource> <!--RusMath2570072Shiryaeva-m2--> </InlineEquation>, the existence of two domains within the unit square in which Grubbs’s copula coincides with the lower Fréchet–Hoeffding bound is proved. In the case of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n = 3\)</EquationSource> <!--RusMath2570072Shiryaeva-m3--> </InlineEquation>, Grubbs’s copula is the Fréchet–Hoeffding lower bound. Grubbs’s copula rotated by 180° also partially coincides with the Fréchet–Hoeffding lower bound (in the case of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n &gt; 3\)</EquationSource> <!--RusMath2570072Shiryaeva-m4--> </InlineEquation>) and is the Fréchet–Hoeffding lower bound (in the case of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(n = 3\)</EquationSource> <!--RusMath2570072Shiryaeva-m5--> </InlineEquation>). We prove that Grubbs’s copulas rotated by 90° and 270° partially coincide with the Fréchet–Hoeffding upper bound (in the case of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(n &gt; 3\)</EquationSource> <!--RusMath2570072Shiryaeva-m6--> </InlineEquation>) and become the Fréchet–Hoeffding upper bound (in the case <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(n = 3\)</EquationSource> <!--RusMath2570072Shiryaeva-m7--> </InlineEquation>).</p>

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On a Family of Copulas Partially Coinciding with the Fréchet–Hoeffding Boundaries

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • L. K. Shiryaeva

摘要

Abstract

Grubbs’s test statistics are studied, that is, absolute values of extreme studentized deviations of \(n\) random observations from the mean. We consider the case when random observations have arbitrary continuous marginal distributions. The existence of two regions is proved; in one of them, the joint distribution function of these statistics is a linear function of their marginal distribution functions, and in the other, the joint distribution function is zero. We construct a Grubbs’s copula from the joint distribution of Grubbs’s statistics. For the case \(n > 3\) , the existence of two domains within the unit square in which Grubbs’s copula coincides with the lower Fréchet–Hoeffding bound is proved. In the case of \(n = 3\) , Grubbs’s copula is the Fréchet–Hoeffding lower bound. Grubbs’s copula rotated by 180° also partially coincides with the Fréchet–Hoeffding lower bound (in the case of \(n > 3\) ) and is the Fréchet–Hoeffding lower bound (in the case of \(n = 3\) ). We prove that Grubbs’s copulas rotated by 90° and 270° partially coincide with the Fréchet–Hoeffding upper bound (in the case of \(n > 3\) ) and become the Fréchet–Hoeffding upper bound (in the case \(n = 3\) ).