scholarly journals Living on the Multidimensional Edge: Seeking Hidden Risks Using Regular Variation

2013 ◽  
Vol 45 (01) ◽  
pp. 139-163 ◽  
Author(s):  
Bikramjit Das ◽  
Abhimanyu Mitra ◽  
Sidney Resnick
Keyword(s):  
Classical Theory ◽  
Regular Variation ◽  
Environmental Science ◽  
Asymptotic Model ◽  
Multivariate Regular Variation ◽  
Tail Risk ◽  
Risk Estimates ◽  
Hidden Regular Variation ◽  
Definition Of

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.

scholarly journals Living on the Multidimensional Edge: Seeking Hidden Risks Using Regular Variation

2013 ◽  
Vol 45 (1) ◽  
pp. 139-163 ◽  
Author(s):  
Bikramjit Das ◽  
Abhimanyu Mitra ◽  
Sidney Resnick
Keyword(s):  
Classical Theory ◽  
Regular Variation ◽  
Environmental Science ◽  
Asymptotic Model ◽  
Multivariate Regular Variation ◽  
Tail Risk ◽  
Risk Estimates ◽  
Hidden Regular Variation ◽  
Definition Of

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.

Relations Between Hidden Regular Variation and the Tail Order of Copulas

2014 ◽  
Vol 51 (01) ◽  
pp. 37-57 ◽  
Author(s):  
Lei Hua ◽  
Harry Joe ◽  
Haijun Li
Keyword(s):  
Integral Representation ◽  
Random Vector ◽  
Regular Variation ◽  
Dependence Structure ◽  
Multivariate Regular Variation ◽  
Intensity Measures ◽  
Extremal Dependence ◽  
Measure Representation ◽  
Hidden Regular Variation ◽  
Type Integral

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.

scholarly journals Relations Between Hidden Regular Variation and the Tail Order of Copulas

2014 ◽  
Vol 51 (1) ◽  
pp. 37-57 ◽  
Author(s):  
Lei Hua ◽  
Harry Joe ◽  
Haijun Li
Keyword(s):  
Integral Representation ◽  
Random Vector ◽  
Regular Variation ◽  
Dependence Structure ◽  
Multivariate Regular Variation ◽  
Intensity Measures ◽  
Extremal Dependence ◽  
Measure Representation ◽  
Hidden Regular Variation ◽  
Type Integral

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.

scholarly journals Characterizations and Examples of Hidden Regular Variation

Extremes ◽  
2004 ◽  
Vol 7 (1) ◽  
pp. 31-67 ◽  
Author(s):  
Krishanu Maulik ◽  
Sidney Resnick
Keyword(s):  
Regular Variation ◽  
Hidden Regular Variation

Subject of Economic Science

2015 ◽  
pp. 115-120
Author(s):  
D. Egorov
Keyword(s):  
Human Behavior ◽  
Classical Theory ◽  
Adam Smith ◽  
A Priori ◽  
Economic Science ◽  
The Public ◽  
Public Consciousness ◽  
Wealth Of Nations ◽  
Definition Of ◽  
The Relationship

Adam Smith defined economics as “the science of the nature and causes of the wealth of nations” (implicitly appealing – in reference to the “wealth” – to the “value”). Neo-classical theory views it as a science “which studies human behavior in terms of the relationship between the objectives and the limited funds that may have a different use of”. The main reason that turns the neo-classical theory (that serves as the now prevailing economic mainstream) into a tool for manipulation of the public consciousness is the lack of measure (elimination of the “value”). Even though the neo-classical definition of the subject of economics does not contain an explicit rejection of objective measures the reference to “human behavior” inevitably implies methodological subjectivism. This makes it necessary to adopt a principle of equilibrium: if you can not objectively (using a solid measurement) compare different states of the system, we can only postulate the existence of an equilibrium point to which the system tends. Neo-classical postulate of equilibrium can not explain the situation non-equilibrium. As a result, the neo-classical theory fails in matching microeconomics to macroeconomics. Moreover, a denial of the category “value” serves as a theoretical basis and an ideological prerequisite of now flourishing manipulative financial technologies. The author believes in the following two principal definitions: (1) economics is a science that studies the economic system, i.e. a system that creates and recombines value; (2) value is a measure of cost of the object. In our opinion, the value is the information cost measure. It should be added that a disclosure of the nature of this category is not an obligatory prerequisite of its introduction: methodologically, it is quite correct to postulate it a priori. The author concludes that the proposed definitions open the way not only to solve the problem of the measurement in economics, but also to address the issue of harmonizing macro- and microeconomics.

scholarly journals A formula for hidden regular variation behavior for symmetric stable distributions

Extremes ◽  
2020 ◽  
Vol 23 (4) ◽  
pp. 667-691
Author(s):  
Malin Palö Forsström ◽  
Jeffrey E. Steif
Keyword(s):  
Power Law ◽  
Spectral Measure ◽  
Regular Variation ◽  
Stable Distributions ◽  
Decay Rates ◽  
Random Vectors ◽  
Power Law Decay ◽  
Exact Power ◽  
Hidden Regular Variation ◽  
Stable Random Vectors

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.

Almost sure limit sets of random samples in ℝd

1988 ◽  
Vol 20 (3) ◽  
pp. 573-599 ◽  
Author(s):  
Richard A. Davis ◽  
Edward Mulrow ◽  
Sidney I. Resnick
Keyword(s):  
Regular Variation ◽  
Almost Sure Convergence ◽  
Random Sets ◽  
Regularity Conditions ◽  
Limit Set ◽  
Multivariate Regular Variation ◽  
Exponential Form ◽  
Limit Sets ◽  
Random Samples ◽  
Distribution Tail

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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