Regular Variation, Conditions of Domain of Attraction and the Existence of the Tail Dependence Function in the General D
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Regular Variation, Conditions of Domain of Attraction and the Existence of the Tail Dependence Function in the General Dependence Case: A Copula Approach Yuri Salazar Flores1 · Adán Díaz Hernández2
© Grace Scientific Publishing 2020
Abstract Given a random vector 𝕏 , Li and Sun (J Appl Probab 46:925–937, 2009), Weng and Zhang (J Multivar Anal 106:178–186, 2012) and Resnick (Extreme values. Regular variation and point processes, Springer, New York, 1987) proved relationships involving the vector being regularly varying, satisfying maximum domain of attraction conditions and the existence of its tail dependence function. Using the corresponding copula function, we give the conditions for the three properties to be equivalent in a general dependence framework. The main contribution of this work is to make well-known results on multivariate domain of attraction, regular variation and tail dependence interchangeable. We show how our results can be applied in a vector of empirical data with heterogeneous marginal distributions. Keywords Regular variation · Maximum domain of attraction · General tail dependence
1 Introduction A univariate distribution function, say Fi , is said to be right tail regularly varying at F (tx) ∞ with heavy-tail index 𝛼 if lim i = t𝛼 for t > 0 . In the multivariate case, conx→∞ F i (x)
sider a d-dimensional positive random vector 𝐗 with right-tail equivalent marginal F (t) distributions, that is, lim i = 1 , for Fi and Fj marginal distributions and t→∞ F j (t)
i, j ∈ {1, … , d} . Following the approach of Li [8], under these conditions, we say 𝐗 is regularly varying with heavy-tail index 𝛼 ≥ 0 , if the univariate margins are right * Yuri Salazar Flores [email protected] 1
Departamento de Matemáticas, Sciences Faculty, National Autonomous University of Mexico (UNAM), Circuito Exterior s/n, Cub. 119, 04510 Mexico City, Mexico
2
Anáhuac University, Huixquilucan, State of Mexico, Mexico
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Journal of Statistical Theory and Practice
(2021) 15:9
tail regularly varying with heavy-tail index 𝛼 and there exists a Radon measure 𝜇 d (finite on compact sets) called the intensity measure, ond ℝ+ �{0} such that = 𝜇(𝐁) for any relatively compact set 𝐁 ⊂ ℝ+ �{0} that satisfies lim P(𝐗∈t𝐁) P(X >t)
t→∞
1
. For all 𝐁 bounded away from 0, the intensity measure 𝜇 satisfies 𝜇(𝜕B) = 0 𝜇(tB) = t−𝛼 𝜇(B) . The assumption of right-tail equivalent marginals is standard in RV theory, and the use of monotone transformations ensures this assumption is satisfied, see, for example, [14, 21]. For the extension to non-positive vectors, refer to Resnick [14], Section 6.5.5, and for a more complete reference of the history of RV theory, see [1]. Now, let F be the distribution function of 𝐗 , and it is said that F is in the maximum domain of attraction (MDA) of multivariate distribution FEV , if there 𝛼i,n > 0 𝛽i,n ∈ ℝ exist constants and for such that n∈ℕ lim F n (𝛼1,n x1 + 𝛽1,n , … , 𝛼d,n xd + 𝛽d,n ) = FEV (x1 , … , xd ) . Given a distribution F, n→∞ according
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