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Artigos de revistas sobre o tema "Multivariate and hidden regular variation"

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Autor: Grafiati

Publicado: 25 de maio de 2024

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1

Heffernan, Janet, e Sidney Resnick. "Hidden regular variation and the rank transform". Advances in Applied Probability 37, n.º 2 (junho de 2005): 393–414. http://dx.doi.org/10.1239/aap/1118858631.

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Random vectors in the positive orthant whose distributions possess hidden regular variation are a subclass of those whose distributions are multivariate regularly varying with asymptotic independence. The concept is an elaboration of the coefficient of tail dependence of Ledford and Tawn. We show that the rank transform that brings unequal marginals to the standard case also preserves the hidden regular variation. We discuss applications of the results to two examples, one involving flood risk and the other Internet data.

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Heffernan, Janet, e Sidney Resnick. "Hidden regular variation and the rank transform". Advances in Applied Probability 37, n.º 02 (junho de 2005): 393–414. http://dx.doi.org/10.1017/s0001867800000239.

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Random vectors in the positive orthant whose distributions possess hidden regular variation are a subclass of those whose distributions are multivariate regularly varying with asymptotic independence. The concept is an elaboration of the coefficient of tail dependence of Ledford and Tawn. We show that the rank transform that brings unequal marginals to the standard case also preserves the hidden regular variation. We discuss applications of the results to two examples, one involving flood risk and the other Internet data.

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3

Resnick, Sidney I. "Multivariate regular variation on cones: application to extreme values, hidden regular variation and conditioned limit laws". Stochastics 80, n.º 2-3 (abril de 2008): 269–98. http://dx.doi.org/10.1080/17442500701830423.

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4

Das, Bikramjit, Abhimanyu Mitra e Sidney Resnick. "Living on the Multidimensional Edge: Seeking Hidden Risks Using Regular Variation". Advances in Applied Probability 45, n.º 1 (março de 2013): 139–63. http://dx.doi.org/10.1239/aap/1363354106.

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Multivariate regular variation plays a role in assessing tail risk in diverse applications such as finance, telecommunications, insurance, and environmental science. The classical theory, being based on an asymptotic model, sometimes leads to inaccurate and useless estimates of probabilities of joint tail regions. This problem can be partly ameliorated by using hidden regular variation (see Resnick (2002) and Mitra and Resnick (2011)). We offer a more flexible definition of hidden regular variation that provides improved risk estimates for a larger class of tail risk regions.

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Das, Bikramjit, Abhimanyu Mitra e Sidney Resnick. "Living on the Multidimensional Edge: Seeking Hidden Risks Using Regular Variation". Advances in Applied Probability 45, n.º 01 (março de 2013): 139–63. http://dx.doi.org/10.1017/s0001867800006224.

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Multivariate regular variation plays a role in assessing tail risk in diverse applications such as finance, telecommunications, insurance, and environmental science. The classical theory, being based on an asymptotic model, sometimes leads to inaccurate and useless estimates of probabilities of joint tail regions. This problem can be partly ameliorated by using hidden regular variation (see Resnick (2002) and Mitra and Resnick (2011)). We offer a more flexible definition of hidden regular variation that provides improved risk estimates for a larger class of tail risk regions.

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6

Hua, Lei, Harry Joe e Haijun Li. "Relations Between Hidden Regular Variation and the Tail Order of Copulas". Journal of Applied Probability 51, n.º 1 (março de 2014): 37–57. http://dx.doi.org/10.1239/jap/1395771412.

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We study the relations between the tail order of copulas and hidden regular variation (HRV) on subcones generated by order statistics. Multivariate regular variation (MRV) and HRV deal with extremal dependence of random vectors with Pareto-like univariate margins. Alternatively, if one uses a copula to model the dependence structure of a random vector then the upper exponent and tail order functions can be used to capture the extremal dependence structure. After defining upper exponent functions on a series of subcones, we establish the relation between the tail order of a copula and the tail indexes for MRV and HRV. We show that upper exponent functions of a copula and intensity measures of MRV/HRV can be represented by each other, and the upper exponent function on subcones can be expressed by a Pickands-type integral representation. Finally, a mixture model is given with the mixing random vector leading to the finite-directional measure in a product-measure representation of HRV intensity measures.

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7

Hua, Lei, Harry Joe e Haijun Li. "Relations Between Hidden Regular Variation and the Tail Order of Copulas". Journal of Applied Probability 51, n.º 01 (março de 2014): 37–57. http://dx.doi.org/10.1017/s0021900200010068.

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We study the relations between the tail order of copulas and hidden regular variation (HRV) on subcones generated by order statistics. Multivariate regular variation (MRV) and HRV deal with extremal dependence of random vectors with Pareto-like univariate margins. Alternatively, if one uses a copula to model the dependence structure of a random vector then the upper exponent and tail order functions can be used to capture the extremal dependence structure. After defining upper exponent functions on a series of subcones, we establish the relation between the tail order of a copula and the tail indexes for MRV and HRV. We show that upper exponent functions of a copula and intensity measures of MRV/HRV can be represented by each other, and the upper exponent function on subcones can be expressed by a Pickands-type integral representation. Finally, a mixture model is given with the mixing random vector leading to the finite-directional measure in a product-measure representation of HRV intensity measures.

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8

Simpson, E. S., J. L. Wadsworth e J. A. Tawn. "Determining the dependence structure of multivariate extremes". Biometrika 107, n.º 3 (7 de maio de 2020): 513–32. http://dx.doi.org/10.1093/biomet/asaa018.

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Summary In multivariate extreme value analysis, the nature of the extremal dependence between variables should be considered when selecting appropriate statistical models. Interest often lies in determining which subsets of variables can take their largest values simultaneously while the others are of smaller order. Our approach to this problem exploits hidden regular variation properties on a collection of nonstandard cones, and provides a new set of indices that reveal aspects of the extremal dependence structure not available through existing measures of dependence. We derive theoretical properties of these indices, demonstrate their utility through a series of examples, and develop methods of inference that also estimate the proportion of extremal mass associated with each cone. We apply the methods to river flows in the U.K., estimating the probabilities of different subsets of sites being large simultaneously.

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9

Mitra, Abhimanyu, e Sidney I. Resnick. "Hidden Regular Variation and Detection of Hidden Risks". Stochastic Models 27, n.º 4 (outubro de 2011): 591–614. http://dx.doi.org/10.1080/15326349.2011.614183.

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10

Maulik, Krishanu, e Sidney Resnick. "Characterizations and Examples of Hidden Regular Variation". Extremes 7, n.º 1 (março de 2004): 31–67. http://dx.doi.org/10.1007/s10687-004-4728-4.

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11

Kim, Moosup. "Simulation of elliptical multivariate regular variation". Journal of the Korean Data And Information Science Society 33, n.º 3 (31 de maio de 2022): 347–57. http://dx.doi.org/10.7465/jkdi.2022.33.3.347.

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12

Yakymiv, A. L. "Multivariate Regular Variation in Probability Theory". Journal of Mathematical Sciences 246, n.º 4 (19 de março de 2020): 580–86. http://dx.doi.org/10.1007/s10958-020-04763-8.

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13

Basrak, Bojan, Richard A. Davis e Thomas Mikosch. "A characterization of multivariate regular variation". Annals of Applied Probability 12, n.º 3 (agosto de 2002): 908–20. http://dx.doi.org/10.1214/aoap/1031863174.

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14

Joe, Harry, e Haijun Li. "Tail Risk of Multivariate Regular Variation". Methodology and Computing in Applied Probability 13, n.º 4 (10 de junho de 2010): 671–93. http://dx.doi.org/10.1007/s11009-010-9183-x.

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15

Das, Bikramjit, e Sidney I. Resnick. "Models with Hidden Regular Variation: Generation and Detection". Stochastic Systems 5, n.º 2 (dezembro de 2015): 195–238. http://dx.doi.org/10.1287/14-ssy141.

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Das, Bikramjit, e Sidney I. Resnick. "Models with hidden regular variation: Generation and detection". Stochastic Systems 5, n.º 2 (2015): 195–238. http://dx.doi.org/10.1214/14-ssy141.

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17

Lindskog, Filip, Sidney I. Resnick e Joyjit Roy. "Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps". Probability Surveys 11 (2014): 270–314. http://dx.doi.org/10.1214/14-ps231.

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18

Eder, Irmingard, e Claudia Klüppelberg. "Pareto Lévy Measures and Multivariate Regular Variation". Advances in Applied Probability 44, n.º 1 (março de 2012): 117–38. http://dx.doi.org/10.1239/aap/1331216647.

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We consider regular variation of a Lévy process X := (Xt)t≥0 in with Lévy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Lévy measures to those of a standard 1-stable Lévy process, we decouple the marginal Lévy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Lévy measure, which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Lévy measures and the Pareto Lévy measure. Moreover, we define upper and lower tail dependence coefficients for the Lévy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Lévy measures.

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19

Eder, Irmingard, e Claudia Klüppelberg. "Pareto Lévy Measures and Multivariate Regular Variation". Advances in Applied Probability 44, n.º 01 (março de 2012): 117–38. http://dx.doi.org/10.1017/s0001867800005474.

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We consider regular variation of a Lévy process X := ( X t) t≥0 in with Lévy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Lévy measures to those of a standard 1-stable Lévy process, we decouple the marginal Lévy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Lévy measure, which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Lévy measures and the Pareto Lévy measure. Moreover, we define upper and lower tail dependence coefficients for the Lévy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Lévy measures.

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20

Meyer, Nicolas, e Olivier Wintenberger. "Sparse regular variation". Advances in Applied Probability 53, n.º 4 (22 de novembro de 2021): 1115–48. http://dx.doi.org/10.1017/apr.2021.14.

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AbstractRegular variation provides a convenient theoretical framework for studying large events. In the multivariate setting, the spectral measure characterizes the dependence structure of the extremes. This measure gathers information on the localization of extreme events and often has sparse support since severe events do not simultaneously occur in all directions. However, it is defined through weak convergence, which does not provide a natural way to capture this sparsity structure. In this paper, we introduce the notion of sparse regular variation, which makes it possible to better learn the dependence structure of extreme events. This concept is based on the Euclidean projection onto the simplex, for which efficient algorithms are known. We prove that under mild assumptions sparse regular variation and regular variation are equivalent notions, and we establish several results for sparsely regularly varying random vectors.

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21

Das, Bikramjit, e Sidney I. Resnick. "Hidden regular variation under full and strong asymptotic dependence". Extremes 20, n.º 4 (17 de março de 2017): 873–904. http://dx.doi.org/10.1007/s10687-017-0290-8.

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22

Resnick, Sidney I., e Joyjit Roy. "Hidden regular variation of moving average processes with heavy-tailed innovations". Journal of Applied Probability 51, A (dezembro de 2014): 267–79. http://dx.doi.org/10.1239/jap/1417528480.

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We look at joint regular variation properties of MA(∞) processes of the form X = (Xk, k ∈ Z), where Xk = ∑j=0∞ψjZk-j and the sequence of random variables (Zi, i ∈ Z) are independent and identically distributed with regularly varying tails. We use the setup of MO-convergence and obtain hidden regular variation properties for X under summability conditions on the constant coefficients (ψj: j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

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23

Resnick, Sidney I., e Joyjit Roy. "Hidden regular variation of moving average processes with heavy-tailed innovations". Journal of Applied Probability 51, A (dezembro de 2014): 267–79. http://dx.doi.org/10.1017/s002190020002132x.

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We look at joint regular variation properties of MA(∞) processes of the form X = (X k , k ∈ Z), where X k = ∑ j=0 ∞ψ j Z k-j and the sequence of random variables (Z i , i ∈ Z) are independent and identically distributed with regularly varying tails. We use the setup of M O -convergence and obtain hidden regular variation properties for X under summability conditions on the constant coefficients (ψ j : j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

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24

Das, Bikramjit, e Vicky Fasen-Hartmann. "Conditional excess risk measures and multivariate regular variation". Statistics & Risk Modeling 36, n.º 1-4 (1 de dezembro de 2019): 1–23. http://dx.doi.org/10.1515/strm-2018-0030.

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Abstract Conditional excess risk measures like Marginal Expected Shortfall and Marginal Mean Excess are designed to aid in quantifying systemic risk or risk contagion in a multivariate setting. In the context of insurance, social networks, and telecommunication, risk factors often tend to be heavy-tailed and thus frequently studied under the paradigm of regular variation. We show that regular variation on different subspaces of the Euclidean space leads to these risk measures exhibiting distinct asymptotic behavior. Furthermore, we elicit connections between regular variation on these subspaces and the behavior of tail copula parameters extending previous work and providing a broad framework for studying such risk measures under multivariate regular variation. We use a variety of examples to exhibit where such computations are practically applicable.

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25

Li, Haijun, e Lei Hua. "Higher order tail densities of copulas and hidden regular variation". Journal of Multivariate Analysis 138 (junho de 2015): 143–55. http://dx.doi.org/10.1016/j.jmva.2014.12.010.

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26

Damek, Ewa, Thomas Mikosch, Jan Rosiński e Gennady Samorodnitsky. "General inverse problems for regular variation". Journal of Applied Probability 51, A (dezembro de 2014): 229–48. http://dx.doi.org/10.1239/jap/1417528478.

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Regular variation of distributional tails is known to be preserved by various linear transformations of some random structures. An inverse problem for regular variation aims at understanding whether the regular variation of a transformed random object is caused by regular variation of components of the original random structure. In this paper we build on previous work, and derive results in the multivariate case and in situations where regular variation is not restricted to one particular direction or quadrant.

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Damek, Ewa, Thomas Mikosch, Jan Rosiński e Gennady Samorodnitsky. "General inverse problems for regular variation". Journal of Applied Probability 51, A (dezembro de 2014): 229–48. http://dx.doi.org/10.1017/s0021900200021306.

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Regular variation of distributional tails is known to be preserved by various linear transformations of some random structures. An inverse problem for regular variation aims at understanding whether the regular variation of a transformed random object is caused by regular variation of components of the original random structure. In this paper we build on previous work, and derive results in the multivariate case and in situations where regular variation is not restricted to one particular direction or quadrant.

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28

Cai, Juan-Juan, John H. J. Einmahl e Laurens de Haan. "Estimation of extreme risk regions under multivariate regular variation". Annals of Statistics 39, n.º 3 (junho de 2011): 1803–26. http://dx.doi.org/10.1214/11-aos891.

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29

Kim, Moosup, e Sangyeol Lee. "Estimation of the tail exponent of multivariate regular variation". Annals of the Institute of Statistical Mathematics 69, n.º 5 (18 de julho de 2016): 945–68. http://dx.doi.org/10.1007/s10463-016-0574-9.

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30

Girard, Stéphane, e Gilles Stupfler. "Extreme geometric quantiles in a multivariate regular variation framework". Extremes 18, n.º 4 (1 de outubro de 2015): 629–63. http://dx.doi.org/10.1007/s10687-015-0226-0.

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31

Kim, Moosup. "Simulation of stock prices based on multivariate regular variation". Journal of the Korean Data And Information Science Society 34, n.º 3 (31 de maio de 2023): 365–75. http://dx.doi.org/10.7465/jkdi.2023.34.3.365.

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32

Forsström, Malin Palö, e Jeffrey E. Steif. "A formula for hidden regular variation behavior for symmetric stable distributions". Extremes 23, n.º 4 (9 de julho de 2020): 667–91. http://dx.doi.org/10.1007/s10687-020-00381-4.

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Abstract We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in “different directions”, illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay.

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33

Hitz, Adrien, e Robin Evans. "One-component regular variation and graphical modeling of extremes". Journal of Applied Probability 53, n.º 3 (setembro de 2016): 733–46. http://dx.doi.org/10.1017/jpr.2016.37.

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AbstractThe problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density. We suggest an approach based on graphical models which is suitable for high-dimensional vectors. We introduce the notion of one-component regular variation to describe a function that is regularly varying in its first component. We extend the representation and Karamata's theorem to one-component regularly varying functions, probability distributions and densities, and explain why these results are fundamental in multivariate extreme-value theory. We then generalize the Hammersley–Clifford theorem to relate asymptotic conditional independence to a factorization of the limiting density, and use it to model multivariate tails.

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34

Alanazi, Fadhah Amer. "A Mixture of Regular Vines for Multiple Dependencies". Journal of Probability and Statistics 2021 (4 de maio de 2021): 1–15. http://dx.doi.org/10.1155/2021/5559518.

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To uncover complex hidden dependency structures among variables, researchers have used a mixture of vine copula constructions. To date, these have been limited to a subclass of regular vine models, the so-called drawable vine, fitting only one type of bivariate copula for all variable pairs. However, the variation of complex hidden correlations from one pair of variables to another is more likely to be present in many real datasets. Single-type bivariate copulas are unable to deal with such a problem. In addition, the regular vine copula model is much more capable and flexible than its subclasses. Hence, to fully uncover and describe complex hidden dependency structures among variables and provide even further flexibility to the mixture of regular vine models, a mixture of regular vine models, with a mixed choice of bivariate copulas, is proposed in this paper. The model was applied to simulated and real data to illustrate its performance. The proposed model shows significant performance over the mixture of R-vine densities with a single copula family fitted to all pairs.

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Maume-Deschamps, Véronique, Didier Rullière e Khalil Said. "Extremes for multivariate expectiles". Statistics & Risk Modeling 35, n.º 3-4 (1 de dezembro de 2018): 111–40. http://dx.doi.org/10.1515/strm-2017-0014.

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Abstract Multivariate expectiles, a new family of vector-valued risk measures, were recently introduced in the literature. [22]. Here we investigate the asymptotic behavior of these measures in a multivariate regular variation context. For models with equivalent tails, we propose an estimator of extreme multivariate expectiles in the Fréchet domain of attraction case with asymptotic independence, or for comonotonic marginal distributions.

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Hult, Henrik, e Filip Lindskog. "On regular variation for infinitely divisible random vectors and additive processes". Advances in Applied Probability 38, n.º 1 (março de 2006): 134–48. http://dx.doi.org/10.1239/aap/1143936144.

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We study the tail behavior of regularly varying infinitely divisible random vectors and additive processes, i.e. stochastic processes with independent but not necessarily stationary increments. We show that the distribution of an infinitely divisible random vector is tail equivalent to its Lévy measure and we study the asymptotic decay of the probability for an additive process to hit sets far away from the origin. The results are extensions of known univariate results to the multivariate setting; we exemplify some of the difficulties that arise in the multivariate case.

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Hult, Henrik, e Filip Lindskog. "On regular variation for infinitely divisible random vectors and additive processes". Advances in Applied Probability 38, n.º 01 (março de 2006): 134–48. http://dx.doi.org/10.1017/s0001867800000847.

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We study the tail behavior of regularly varying infinitely divisible random vectors and additive processes, i.e. stochastic processes with independent but not necessarily stationary increments. We show that the distribution of an infinitely divisible random vector is tail equivalent to its Lévy measure and we study the asymptotic decay of the probability for an additive process to hit sets far away from the origin. The results are extensions of known univariate results to the multivariate setting; we exemplify some of the difficulties that arise in the multivariate case.

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Moser, Martin, e Robert Stelzer. "Functional regular variation of Lévy-driven multivariate mixed moving average processes". Extremes 16, n.º 3 (7 de fevereiro de 2013): 351–82. http://dx.doi.org/10.1007/s10687-012-0165-y.

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Weller, G. B., e D. Cooley. "A sum characterization of hidden regular variation with likelihood inference via expectation-maximization". Biometrika 101, n.º 1 (6 de fevereiro de 2014): 17–36. http://dx.doi.org/10.1093/biomet/ast046.

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Mansouri, Sadollah, Masood Soltani Najafabadi, Maghsadollah Esmailov e Mostafa Aghaee. "Functional Factor Analysis In Sesame Under Water - Limiting Stress: New Concept On An Old Method". Plant Breeding and Seed Science 70, n.º 1 (1 de dezembro de 2014): 91–104. http://dx.doi.org/10.1515/plass-2015-0016.

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Abstract Multivariate statistical analysis, through their ability to extract hidden relationship between various traits, has a wide application in breeding programs. Having physiological concept on the multivariate analysis, factor analysis was used to extract differential relationships between different components involving in assimilate partitioning in sesame under regular irrigation regime and limited irrigation. The analysis revealed that under regular irrigation regime, the stored and/or currently produced assimilates are allocated to the filling seeds. However, incidence of water shortage in the beginning of flowering time make shifts in assimilate partitioning from formation of new seeds or capsules to the not-matured pre-formed seeds, which results in seeds with more nutrient storage. This indicates the requirement for change in breeding strategies under sub-optimal condition. The possible common language between factor concept in multivariate analysis, QTLs in genetics, and transcription factors in molecular biology is indicated.

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Ho, Zhen Wai Olivier, e Clément Dombry. "Simple models for multivariate regular variation and the Hüsler–Reiß Pareto distribution". Journal of Multivariate Analysis 173 (setembro de 2019): 525–50. http://dx.doi.org/10.1016/j.jmva.2019.04.008.

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42

Davis, Richard A., Edward Mulrow e Sidney I. Resnick. "Almost sure limit sets of random samples in ℝd". Advances in Applied Probability 20, n.º 3 (setembro de 1988): 573–99. http://dx.doi.org/10.2307/1427036.

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If {Xj, } is a sequence of i.i.d. random vectors in , when do there exist scaling constants bn > 0 such that the sequence of random sets converges almost surely in the space of compact subsets of to a limit set? A multivariate regular variation condition on a properly defined distribution tail guarantees the almost sure convergence but without certain regularity conditions surprises can occur. When a density exists, an exponential form of regular variation plus some regularity guarantees the convergence.

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Davis, Richard A., Edward Mulrow e Sidney I. Resnick. "Almost sure limit sets of random samples in ℝd". Advances in Applied Probability 20, n.º 03 (setembro de 1988): 573–99. http://dx.doi.org/10.1017/s0001867800018152.

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If {Xj , } is a sequence of i.i.d. random vectors in , when do there exist scaling constants bn > 0 such that the sequence of random sets converges almost surely in the space of compact subsets of to a limit set? A multivariate regular variation condition on a properly defined distribution tail guarantees the almost sure convergence but without certain regularity conditions surprises can occur. When a density exists, an exponential form of regular variation plus some regularity guarantees the convergence.

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44

Li, Haijun, e Yannan Sun. "Tail Dependence for Heavy-Tailed Scale Mixtures of Multivariate Distributions". Journal of Applied Probability 46, n.º 4 (dezembro de 2009): 925–37. http://dx.doi.org/10.1239/jap/1261670680.

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The tail dependence of multivariate distributions is frequently studied via the tool of copulas. In this paper we develop a general method, which is based on multivariate regular variation, to evaluate the tail dependence of heavy-tailed scale mixtures of multivariate distributions, whose copulas are not explicitly accessible. Tractable formulae for tail dependence parameters are derived, and a sufficient condition under which the parameters are monotone with respect to the heavy tail index is obtained. The multivariate elliptical distributions are discussed to illustrate the results.

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Li, Haijun, e Yannan Sun. "Tail Dependence for Heavy-Tailed Scale Mixtures of Multivariate Distributions". Journal of Applied Probability 46, n.º 04 (dezembro de 2009): 925–37. http://dx.doi.org/10.1017/s0021900200006057.

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The tail dependence of multivariate distributions is frequently studied via the tool of copulas. In this paper we develop a general method, which is based on multivariate regular variation, to evaluate the tail dependence of heavy-tailed scale mixtures of multivariate distributions, whose copulas are not explicitly accessible. Tractable formulae for tail dependence parameters are derived, and a sufficient condition under which the parameters are monotone with respect to the heavy tail index is obtained. The multivariate elliptical distributions are discussed to illustrate the results.

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Wang, Tiandong, e Sidney I. Resnick. "Multivariate Regular Variation of Discrete Mass Functions with Applications to Preferential Attachment Networks". Methodology and Computing in Applied Probability 20, n.º 3 (2 de junho de 2016): 1029–42. http://dx.doi.org/10.1007/s11009-016-9503-x.

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Sundravel, K. Vijaya, S. Ramesh e D. Jegatheeswaran. "Design and formulation of microbially induced self-healing concrete for building structure strength enhancement". Materials Express 11, n.º 11 (1 de novembro de 2021): 1753–65. http://dx.doi.org/10.1166/mex.2021.2104.

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Self-healing concrete is described as the capability of material to repair their cracks independently. Cracks in concrete are well-known circumstance because of their short tensile strength. Many researchers carried out their research on self-healing concrete using different classification and clustering methods. But the temperature variation and pH variation were not minimized. In order to address these problems, a Multivariate Logistic Regressed Chi-Square Deep Recurrent Neural Network based Self-Healing (MLRCSDRNN-SH) Method is introduced. The main aim of MLRCSDRNN-SH method is to improve building structures strength through inducing the micro-bacteria in concrete. Multiple Logistic Regressed Chi-Square Deep Recurrent Neural Network (MLRCSDRNN) is used to revise bacteria’s stress-strain behaviour towards enhanced material strength in the MLRCSDRNN-SH approach. Initially, the bacteria selection is carried out in alkaline environment like Bacillus subtilis, E. coli and Pseudomonas sps. The data sample is given to the input layer. The input layer transmits sample to the hidden layer 1. The regression analysis is carried out between the multiple independent variables (i.e., parameters) using multivariate logistic function for improving the building structure strength. The regressed value is transmitted to the hidden layer 2. The pearson chi-squared independence hypothesis is performed to identify the probability of crack self-healing property for increasing the building structure strength. When probability value is higher, then the building structure strength is high. Otherwise, the output of second hidden layer is feedback to the input of hidden layer 1. The mixture with higher strength of building structure is sent to the output layer. Several specimens have different sizes used by various researchers for bacterial material study in comparison with the concrete. Depending on experimental results, compressive strength restoration proved higher self-healing ability of the concrete.

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Tang, Qihe, e Zhongyi Yuan. "Asymptotic Analysis of the Loss Given Default in the Presence of Multivariate Regular Variation". North American Actuarial Journal 17, n.º 3 (3 de julho de 2013): 253–71. http://dx.doi.org/10.1080/10920277.2013.830557.

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Xing, Guo-dong, e Xiaoli Gan. "Asymptotic analysis of tail distortion risk measure under the framework of multivariate regular variation". Communications in Statistics - Theory and Methods 49, n.º 12 (12 de março de 2019): 2931–41. http://dx.doi.org/10.1080/03610926.2019.1584312.

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Glass, I. S. "3.14. Infrared variations of active galaxies: what they tell us". Symposium - International Astronomical Union 184 (1998): 117–18. http://dx.doi.org/10.1017/s0074180900084278.

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Little is known about the long-term infrared variations of Seyfert galaxies. Are they large or small, fast or slow, regular or irregular? Do they possess variable components hidden at visible wavelengths? Can their variational time-scales give us information about activity on sub-milliarcsecond spatial scales? Do the infrared measurements show well-defined flux variation gradients? Is the infrared flux delayed with respect to changes in the output of the central engine? And do the infrared variability studies support the “Unified Model”?

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