Hidden Regular Variation and Detection of Hidden Risks

Mitra, Abhimanyu; Resnick, Sidney I.

doi:10.1080/15326349.2011.614183

Cited by 25 publications

(70 citation statements)

References 34 publications

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“…2.4 and then use the properties of HRV to evaluate the asymptotic expansions of P(S n > t, N t = 2). For the case when n = 2, the result can be verified easily by the property of HRV (see Mitra and Resnick (Mitra and Resnick 2010)). …”

Section: Lemmasmentioning

confidence: 60%

Second-order properties of tail probabilities of sums and randomly weighted sums

Mao

2015

Extremes

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Let X 1 , . . . , X n be independent nonnegative random variables with respective survival functions F 1 , . . . , F n , and let 1 , . . . , n be (not necessarily independent) nonnegative random variables, independent of X 1 , . . . , X n , satisfying certain moment conditions. This paper consists of two parts. In the first part, we investigate second-order expansions of P n i=1 X i > t as t → ∞ under the assumption that the F i are of second-order regular variation (2RV) with the same first-order index but with different second-order indexes. In the second part, under the assumption that the F 1 = · · · = F n have 2RV tails, second-order expansions of tail probabilities of the randomly weighted sum n i=1 i X i are studied. The closure property of 2RV under randomly weighted sum is also discussed. The main results in this paper generalize and strengthen several known results in the literature.

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Section: Lemmasmentioning

confidence: 60%

Second-order properties of tail probabilities of sums and randomly weighted sums

Mao

2015

Extremes

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“…. , d. In contrast to the previous study on norm-based polar transforms for MRV on subcones, the method in Mitra and Resnick (2011) fixes directions on an order-statisticsbased unit envelope δN l := {x ∈ E (l) : x [l] = 1} that wraps all open portions of the boundaries of subcone E (l) from within. Note that δN l is always compact within E (l) , and this leads to a product-measure representation for the intensity measure ν l (·) of MRV on E (l) , where the spectral or directional measure S l (·) is always finite.…”

Section: Remark 33 the Proof Of Proposition 33 Also Yields An Intementioning

confidence: 97%

“…, d. Precisely speaking, the MRV discussed in Section 2 is the MRV on the cone E (1) . MRV can also be defined on subcones (see Mitra and Resnick (2010), (2011)). Let X be a nonnegative random vector.…”

Section: Tail Order For Hrv On Subconesmentioning

confidence: 99%

“…For a subspace E ⊆R d + , let M + (E) denote the class of the nonnegative Radon measures on E. Mitra and Resnick (2011) provided a representation of MRV on a general subcone E (l) for l = 1, . .…”

Section: Remark 33 the Proof Of Proposition 33 Also Yields An Intementioning

confidence: 99%

“…Recently, Mitra and Resnick (2010), (2011) employed order statistics to construct a productmeasure decomposition for characterizing HRV. Their approach overcomes the issue of infinite angular measures for HRV, studied earlier in Maulik and Resnick (2004) or Section 9.4.1 of Resnick (2007).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Relations Between Hidden Regular Variation and the Tail Order of Copulas

Hua¹,

Joe²,

Li³

2014

J. Appl. Probab.

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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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Foundation

Resnick

2024

The Art of Finding Hidden Risks

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Hidden Regular Variation and Detection of Hidden Risks

Cited by 25 publications

References 34 publications

Second-order properties of tail probabilities of sums and randomly weighted sums

Second-order properties of tail probabilities of sums and randomly weighted sums

Relations Between Hidden Regular Variation and the Tail Order of Copulas

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