scholarly journals Models with Hidden Regular Variation: Generation and Detection

2015 ◽  
Vol 5 (2) ◽  
pp. 195-238 ◽  
Author(s):  
Bikramjit Das ◽  
Sidney I. Resnick
Keyword(s):  
Regular Variation ◽  
Hidden Regular Variation

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

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.

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 Hidden regular variation of moving average processes with heavy-tailed innovations

2014 ◽  
Vol 51 (A) ◽  
pp. 267-279 ◽  
Author(s):  
Sidney I. Resnick ◽  
Joyjit Roy
Keyword(s):  
Sequence Space ◽  
Regular Variation ◽  
Moving Average ◽  
Random Variables ◽  
Continuity Properties ◽  
Hidden Regular Variation ◽  
Constant Coefficients ◽  
Moving Average Processes ◽  
Regularly Varying ◽  
Heavy Tailed

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.

scholarly journals Models with hidden regular variation: Generation and detection

2015 ◽  
Vol 5 (2) ◽  
pp. 195-238 ◽  
Author(s):  
Bikramjit Das ◽  
Sidney I. Resnick
Keyword(s):  
Regular Variation ◽  
Hidden Regular Variation

scholarly journals Hidden regular variation of moving average processes with heavy-tailed innovations

2014 ◽  
Vol 51 (A) ◽  
pp. 267-279
Author(s):  
Sidney I. Resnick ◽  
Joyjit Roy
Keyword(s):  
Sequence Space ◽  
Regular Variation ◽  
Moving Average ◽  
Random Variables ◽  
Continuity Properties ◽  
Hidden Regular Variation ◽  
Constant Coefficients ◽  
Moving Average Processes ◽  
Regularly Varying ◽  
Heavy Tailed

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.

Hidden Regular Variation and Detection of Hidden Risks

2011 ◽  
Vol 27 (4) ◽  
pp. 591-614 ◽  
Author(s):  
Abhimanyu Mitra ◽  
Sidney I. Resnick
Keyword(s):  
Regular Variation ◽  
Hidden Regular Variation

scholarly journals Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps

2014 ◽  
Vol 11 (0) ◽  
pp. 270-314 ◽  
Author(s):  
Filip Lindskog ◽  
Sidney I. Resnick ◽  
Joyjit Roy
Keyword(s):  
Regular Variation ◽  
Metric Spaces ◽  
Hidden Regular Variation ◽  
Regularly Varying

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.

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