Academic literature on the topic 'Multivariate and hidden regular variation'
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Journal articles on the topic "Multivariate and hidden regular variation"
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Heffernan, Janet, and Sidney Resnick. "Hidden regular variation and the rank transform." Advances in Applied Probability 37, no. 2 (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, and Sidney Resnick. "Hidden regular variation and the rank transform." Advances in Applied Probability 37, no. 02 (2005): 393–414. http://dx.doi.org/10.1017/s0001867800000239.
Full textAbstract:
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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Resnick, Sidney I. "Multivariate regular variation on cones: application to extreme values, hidden regular variation and conditioned limit laws." Stochastics 80, no. 2-3 (2008): 269–98. http://dx.doi.org/10.1080/17442500701830423.
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Das, Bikramjit, Abhimanyu Mitra, and Sidney Resnick. "Living on the Multidimensional Edge: Seeking Hidden Risks Using Regular Variation." Advances in Applied Probability 45, no. 1 (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, and Sidney Resnick. "Living on the Multidimensional Edge: Seeking Hidden Risks Using Regular Variation." Advances in Applied Probability 45, no. 01 (2013): 139–63. http://dx.doi.org/10.1017/s0001867800006224.
Full textAbstract:
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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Hua, Lei, Harry Joe, and Haijun Li. "Relations Between Hidden Regular Variation and the Tail Order of Copulas." Journal of Applied Probability 51, no. 1 (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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Hua, Lei, Harry Joe, and Haijun Li. "Relations Between Hidden Regular Variation and the Tail Order of Copulas." Journal of Applied Probability 51, no. 01 (2014): 37–57. http://dx.doi.org/10.1017/s0021900200010068.
Full textAbstract:
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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Simpson, E. S., J. L. Wadsworth, and J. A. Tawn. "Determining the dependence structure of multivariate extremes." Biometrika 107, no. 3 (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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Mitra, Abhimanyu, and Sidney I. Resnick. "Hidden Regular Variation and Detection of Hidden Risks." Stochastic Models 27, no. 4 (2011): 591–614. http://dx.doi.org/10.1080/15326349.2011.614183.
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Maulik, Krishanu, and Sidney Resnick. "Characterizations and Examples of Hidden Regular Variation." Extremes 7, no. 1 (2004): 31–67. http://dx.doi.org/10.1007/s10687-004-4728-4.
Full textDissertations / Theses on the topic "Multivariate and hidden regular variation"
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Legrand, Juliette. "Simulation and assessment of multivariate extreme models for environmental data." Electronic Thesis or Diss., université Paris-Saclay, 2022. http://www.theses.fr/2022UPASJ015.
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L'estimation précise des probabilités d'occurrence des événements extrêmes environnementaux est une préoccupation majeure dans l'évaluation des risques. Pour l'ingénierie côtière par exemple, le dimensionnement de structures implantées sur ou à proximité des côtes doit être tel qu'elles résistent aux événements les plus sévères qu'elles puissent rencontrer au cours de leur vie. Cette thèse porte sur la simulation d'événements extrêmes multivariés, motivée par des applications aux hauteurs significatives de vagues, et sur l'évaluation de modèles de prédiction d'occurrence d'événements extrêmes.Dans la première partie du manuscrit, nous proposons et étudions un simulateur stochastique qui génère conjointement, en fonction de certaines conditions d'état de mer au large, des extrêmes de hauteur significative de vagues (Hs) au large et à la côte. Pour cela, nous nous appuyons sur l'approche par dépassements de seuils bivariés et nous développons un algorithme de simulation non-paramétrique de lois de Pareto généralisées bivariées. À partir de ce simulateur d'événements cooccurrents, nous dérivons un modèle de simulation conditionnel. Les deux algorithmes de simulation sont mis en oeuvre sur des expériences numériques et appliqués aux extrêmes de Hs près des côtes bretonnes françaises. Un autre développement est traité quant à la modélisation des lois marginales des Hs. Afin de prendre en compte leur non-stationnaritée, nous adaptons une extension de la loi de Pareto généralisée, en considérant l'effet de la période et de la direction pic sur ses paramètres.La deuxième partie de cette thèse apporte un développement plus théorique. Pour évaluer différents modèles de prédiction d'extrêmes, nous étudions le cas spécifique des classifieurs binaires, qui constituent la forme la plus simple de prévision et de processus décisionnel : un événement extrême s'est produit ou ne s'est pas produit. Des fonctions de risque adaptées à la classification binaire d'événements extrêmes sont développées, ce qui nous permet de répondre à notre deuxième question. Leurs propriétés sont établies dans le cadre de la variation régulière multivariée et de la variation régulière cachée, permettant de considérer des formes plus fines d'indépendance asymptotique. Ces développements sont ensuite appliqués aux débits de rivière extrêmes<br>Accurate estimation of the occurrence probabilities of extreme environmental events is a major issue for risk assessment. For example, in coastal engineering, the design of structures installed at or near the coasts must be such that they can withstand the most severe events they may encounter in their lifetime. This thesis focuses on the simulation of multivariate extremes, motivated by applications to significant wave height, and on the evaluation of models predicting the occurrences of extreme events.In the first part of the manuscript, we propose and study a stochastic simulator that, given offshore conditions, produces jointly offshore and coastal extreme significant wave heights (Hs). We rely on bivariate Peaks over Threshold and develop a non-parametric simulation scheme of bivariate generalised Pareto distributions. From such joint simulator, we derive a conditional simulation model. Both simulation algorithms are applied to numerical experiments and to extreme Hs near the French Brittanny coast. A further development is addressed regarding the marginal modelling of Hs. To take into account non-stationarities, we adapt the extended generalised Pareto model, letting the marginal parameters vary with the peak period and the peak direction.The second part of this thesis provides a more theoretical development. To evaluate different prediction models for extremes, we study the specific case of binary classifiers, which are the simplest type of forecasting and decision-making situation: an extreme event did or did not occur. Risk functions adapted to binary classifiers of extreme events are developed, answering our second question. Their properties are derived under the framework of multivariate regular variation and hidden regular variation, allowing to handle finer types of asymptotic independence. This framework is applied to extreme river discharges
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Mariko, Dioulde Habibatou. "Multivariate Regular Variation and its Applications." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/32756.
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In this thesis, we review the basic notions related to univariate regular variation and study some fundamental properties of regularly varying random variables. We then consider the notion of regular variation in the multivariate case. After collecting some results from multivariate regular variation for random vectors with values in $\mathbb{R}_{+}^{d}$, we discuss its properties and examine several examples of multivariate regularly varying random vectors such as independent and identically distributed random vectors, fully dependent random vectors and other models. We also present the elements of univariate and multivariate extreme value theory and emphasize the connection with multivariate regular variation. Some measures of extreme dependence such as the stable tail dependence function and the Pickands dependence function are presented. We end the study by conducting a data analysis using financial data. In the univariate case, graphical tools such as quantile-quantile plots, mean excess plots and Hill plots are used in order to determine the underlying distribution of the univariate data. In the multivariate case, non-parametric estimators of the stable tail dependence function and the Pickands dependence function are used to describe the dependence structure of the multivariate data.
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Tong, Zhigang. "Statistical Inference for Heavy Tailed Time Series and Vectors." Thesis, Université d'Ottawa / University of Ottawa, 2017. http://hdl.handle.net/10393/35649.
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In this thesis we deal with statistical inference related to extreme value phenomena.
Specifically, if X is a random vector with values in d-dimensional space, our goal is
to estimate moments of ψ(X) for a suitably chosen function ψ when the magnitude
of X is big. We employ the powerful tool of regular variation for random variables,
random vectors and time series to formally define the limiting quantities of interests
and construct the estimators. We focus on three statistical estimation problems: (i)
multivariate tail estimation for regularly varying random vectors, (ii) extremogram
estimation for regularly varying time series, (iii) estimation of the expected shortfall
given an extreme component under a conditional extreme value model. We establish asymptotic normality of estimators for each of the estimation problems. The theoretical findings are supported by simulation studies and the estimation procedures are applied to some financial data.
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Yuan, Zhongyi. "Quantitative analysis of extreme risks in insurance and finance." Diss., University of Iowa, 2013. https://ir.uiowa.edu/etd/2422.
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In this thesis, we aim at a quantitative understanding of extreme risks. We use heavy-tailed distribution functions to model extreme risks, and use various tools, such as copulas and MRV, to model dependence structures. We focus on modeling as well as quantitatively estimating certain measurements of extreme risks.
We start with a credit risk management problem. More specifically, we consider a credit portfolio of multiple obligors subject to possible default. We propose a new structural model for the loss given default, which takes into account the severity of default. Then we study the tail behavior of the loss given default under the assumption that the losses of the obligors jointly follow an MRV structure. This structure provides an ideal framework for modeling both heavy tails and asymptotic dependence. Using HRV, we also accommodate the asymptotically independent case. Multivariate models involving Archimedean copulas, mixtures and linear transforms are revisited.
We then derive asymptotic estimates for the Value at Risk and Conditional Tail Expectation of the loss given default and compare them with the traditional empirical estimates.
Next, we consider an investor who invests in multiple lines of business and study a capital allocation problem. A randomly weighted sum structure is proposed, which can capture both the heavy-tailedness of losses and the dependence among them, while at the same time separates the magnitudes from dependence. To pursue as much generality as possible, we do not impose any requirement on the dependence structure of the random weights. We first study the tail behavior of the total loss and obtain asymptotic formulas under various sets of conditions. Then we derive asymptotic formulas for capital allocation and further refine them to be explicit for some cases.
Finally, we conduct extreme risk analysis for an insurer who makes investments. We consider a discrete-time risk model in which the insurer is allowed to invest a proportion of its wealth in a risky stock and keep the rest in a risk-free bond. Assume that the claim amounts within individual periods follow an autoregressive process with heavy-tailed innovations and that the log-returns of the stock follow another autoregressive process, independent of the former one. We derive an asymptotic formula for the finite-time ruin probability and propose a hybrid method, combining simulation with asymptotics, to compute this ruin probability more efficiently. As an application, we consider a portfolio optimization problem in which we determine the proportion invested in the risky stock that maximizes the expected terminal wealth subject to a constraint on the ruin probability.
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Hitz, Adrien. "Modelling of extremes." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:ad32f298-b140-4aae-b50e-931259714085.
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This work focuses on statistical methods to understand how frequently rare events occur and what the magnitude of extreme values such as large losses is. It lies in a field called extreme value analysis whose scope is to provide support for scientific decision making when extreme observations are of particular importance such as in environmental applications, insurance and finance. In the univariate case, I propose new techniques to model tails of discrete distributions and illustrate them in an application on word frequency and multiple birth data. Suitably rescaled, the limiting tails of some discrete distributions are shown to converge to a discrete generalized Pareto distribution and generalized Zipf distribution respectively. In the multivariate high-dimensional case, I suggest modeling tail dependence between random variables by a graph such that its nodes correspond to the variables and shocks propagate through the edges. Relying on the ideas of graphical models, I prove that if the variables satisfy a new notion called asymptotic conditional independence, then the density of the joint distribution can be simplified and expressed in terms of lower dimensional functions. This generalizes the Hammersley- Clifford theorem and enables us to infer tail distributions from observations in reduced dimension. As an illustration, extreme river flows are modeled by a tree graphical model whose structure appears to recover almost exactly the actual river network. A fundamental concept when studying limiting tail distributions is regular variation. I propose a new notion in the multivariate case called one-component regular variation, of which Karamata's and the representation theorem, two important results in the univariate case, are generalizations. Eventually, I turn my attention to website visit data and fit a censored copula Gaussian graphical model allowing the visualization of users' behavior by a graph.
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Eder, Irmingard [Verfasser]. "First passage events and multivariate regular variation for dependent Lévy processes with applications in insurance / Irmingard Marianne Margarethe Eder." 2009. http://d-nb.info/996373233/34.
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Janßen, Anja. "Über Zusammenhänge von leichten Tails, regulärer Variation und Extremwerttheorie." Doctoral thesis, 2010. http://hdl.handle.net/11858/00-1735-0000-0006-B69F-1.
Full textBooks on the topic "Multivariate and hidden regular variation"
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Omey, E. Multivariate regular variation and applications in probability theory. Economische Hogeschool Sint-Aloysius, 1989.
Find full textBook chapters on the topic "Multivariate and hidden regular variation"
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Resnick, Sidney I. "Multivariate Extremes." In Extreme Values, Regular Variation and Point Processes. Springer New York, 1987. http://dx.doi.org/10.1007/978-0-387-75953-1_6.
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Li, Haijun. "Toward a Copula Theory for Multivariate Regular Variation." In Copulae in Mathematical and Quantitative Finance. Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-35407-6_9.
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"Multivariate Regular Variation." In Inference for Heavy-Tailed Data Analysis. Elsevier, 2017. http://dx.doi.org/10.1016/b978-0-12-804676-0.00004-3.
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"Multivariate regular variation." In Risk Theory. WORLD SCIENTIFIC, 2017. http://dx.doi.org/10.1142/9789813223158_0013.
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Rangeti, Innocent, and Bloodless Dzwairo. "Interpretation of Water Quality Data in uMngeni Basin (South Africa) Using Multivariate Techniques." In River Basin Management - Sustainability Issues and Planning Strategies. IntechOpen, 2021. http://dx.doi.org/10.5772/intechopen.94845.
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The major challenge with regular water quality monitoring programmes is making sense of the large and complex physico-chemical data-sets that are generated in a comparatively short period of time. Consequentially, this presents difficulties for water management practitioners who are expected to make informed decisions based on information extracted from the large data-sets. In addition, the nonlinear nature of water quality data-sets often makes it difficult to interpret the spatio-temporal variations. These reasons necessitated the need for effective methods of interpreting water quality results and drawing meaningful conclusions. Hence, this study applied multivariate techniques, namely Cluster Analysis and Principal Component Analysis, to interpret eight-year (2005–2012) water quality data that was generated from a monitoring exercise at six stations in uMngeni Basin, South Africa. The principal components extracted with eigenvalues of greater than 1 were interpreted while considering the pollution issues in the basin. These extracted components explain 67–76% of the water quality variation among the stations. The derived significant parameters suggest that uMngeni Basin was mainly affected by the catchment’s geological processes, surface runoff, domestic sewage effluent, seasonal variation and agricultural waste. Cluster Analysis grouped the sampling six stations into two clusters namely heavy (B) or low (A), based on the degree of pollution. Cluster A mainly consists of water sampling stations that were located in the outflow of the dam (NDO, IDO, MDO and NDI) and its water can be described as of fairly good quality due to dam retention and attenuation effects. Cluster B mainly consist of dam inflow water sampling stations (MDI and IDI), which can be described as polluted if compared to cluster A. The poor quality water observed at Cluster B sampling stations could be attributed to natural and anthropogenic activities through point source and runoff. The findings could assist in determining an appropriate set of water quality parameters that would indicate variation of water quality in the basin, with minimum loss of information. It is, therefore, recommended that this approach be used to assist decision-makers regarding strategies for minimising catchment pollution.
Conference papers on the topic "Multivariate and hidden regular variation"
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OLUWAJIRE, OLUWATIMILEHIN, KATHERINE BERKOWITZ, and LANDON GRACE. "A PERFORMANCE COMPARISON OF LOW-COST NIR NANO TO NIR MICROPHAZIR FOR POLYMER COMPOSITE CHARACTERIZATION." In Proceedings for the American Society for Composites-Thirty Eighth Technical Conference. Destech Publications, Inc., 2023. http://dx.doi.org/10.12783/asc38/36559.
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Polymer composites are increasingly adopted to complement or replace their metal counterparts used in aerospace, wind turbine, automotive, and other safety-critical structures. However, these structures are susceptible to the development of undetected sub-surface damage, such as low velocity impact damage (LVI), that limits their performance and utility in certain cases. The ability to detect and monitor the progression of this type of damage from the micron level prior to growth to the macroscale is critical to ensuring the long-term safety and reliability of these structures. Near Infrared Spectroscopy (NIRS) is a promising technique for detecting LVI damage because it can probe the internal structure of the material and identify subtle changes in its properties. This study compared the performance of a low-cost NIR Nano (a Texas Instruments NIR Nano evaluation model) to a commercial NIR Microphazir instrument for the detection of LVI damage in glass fiber reinforced polymer composites. We evaluated the two instruments for their ability to detect and measure different levels of impact damage at increasing moisture absorbed by weight. Forty-eight fiberglass laminates consisting of e-glass and epoxy were subjected to either 0J (no damage), 1J, 1.5J, and 2J impact energy using a drop tower outfitted with a 9mm radius hemispherical striker tip impactor. Spectral scans were collected between wavelengths of 900-1700 nm (NIR Nano) and 1600-2400 nm (Microphazir) for all samples prior to moisture absorption. Following moisture contamination, spectral scans were taken at regular intervals of gravimetric moisture gain from 0.05% to 0.15% by weight. Multivariate data analysis methods were used to assess the spatial variation of the absorbance parameter at various amounts of absorbed moisture. The results and discussion emphasize the importance of rigorous calibration and technique selection for reliable and accurate NIR spectroscopy investigation in polymer composites, especially in situations where mobility,