Best-possible bounds for the distribution of a sum — a problem of Kolmogorov

Frank, Michael; Nelsen, Roger B.; Schweizer, B.

doi:10.1007/bf00569989

Cited by 171 publications

(131 citation statements)

References 3 publications

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“…This result was shown by Frank et al (1987), and by Rüschendorf (1982) when n = 2 and C L = W. The copulas C α have recently been investigated by Embrechts et al (2005), who provide many interesting graphical interpretations.…”

Section: Bounds When the Marginal Distributions Are Knownmentioning

confidence: 76%

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Bounds on the value-at-risk for the sum of possibly dependent risks

Mesfioui

Quessy

2005

Insurance: Mathematics and Economics

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In this paper, explicit lower and upper bounds on the value-at-risk (VaR) for the sum of possibly dependent risks are derived when only partial information is available about the dependence structure and the individual behaviors. When the marginal distributions are known, a reformulation of a result of Embrechts et al. [Finan. Stoch. 7 (2003) 145-167] makes it possible, under some regularity conditions, to compute explicit bounds for the VaR under various dependence scenarios. In the case where only the means and the variances of the risks are available, explicit bounds are obtained from an optimization over all possible values of the correlation matrix associated with the vector of risks. Analytical and numerical investigations are presented in order to investigate the quality of these bounds.

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Section: Bounds When the Marginal Distributions Are Knownmentioning

confidence: 76%

“…Frank et al (1987) proved the best-possible nature of these bounds. Williamson and Downs (1990) translated these results into bounds for the value-at-risk of the sum of two risks using the duality principle.…”

Section: Bounds When the Marginal Distributions Are Knownmentioning

confidence: 94%

Bounds on the value-at-risk for the sum of possibly dependent risks

Mesfioui

Quessy

2005

Insurance: Mathematics and Economics

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“…For this purpose, Frank, Nelsen et Schweizer results [16] can be used. Indeed, they provide bounds on the distribution function from Fréchet bounds on the joint distribution [17].…”

Section: Unknown Dependencymentioning

confidence: 99%

Possibility transformation of the sum of two symmetric unimodal independent/comonotone random variables

Mauris¹

2013

Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology

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The paper extends author's previous works on a probability/possibility transformation based on a maximum specificity principle to the case of the sum of two identical unimodal symmetric random variables. This transformation requires the knowledge of the dependency relationship between the two added variables. In fact, the comonotone case is closely related to the Zadeh's extension principle. It often leads to the worst case in terms of specificity of the corresponding possibility distribution, but it may arise that the independent case is worse than the comonotone case, e.g. for symmetric Pareto probability distributions. When no knowledge about the dependence is available, a least specific possibility distribution can be obtained from Fréchet bounds.

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“…A (two-dimensional) copula is a mapping C from the unit square [0,1 ]2 onto the unit interval [0,1] Alternatively, a copula is a bivariate distribution function whose support is contained in the unit square and whose margins are uniform on [0,1].…”

Section: Preliminariesmentioning

confidence: 99%

“…Frank and the authors used copulas and their properties to obtain bestpossible bounds for the distribution function of the sum X+Y of two random variables X and Y whose individual distribution functions F X and Fy are fixed. Our a;m in this paper is to apply the techniques de-veloped in [1] to obtain best-possible bounds for the distribution function of the sum of squares X2+y 2 and for the so-called radial error (X2+y2) 1 have a common distribution function which is concave on (0,oo). We also obtain slmxlar restllts for the radial error and consider the important special case when the margins are normal.…”

Section: Introductionmentioning

confidence: 99%

Bounds for distribution functions of sums of squares and radial errors

Nelsen

Schweizer

1991

International Journal of Mathematics and Mathematical Sciences

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ABSTRACT. Bounds are found for the distribution function of the sum of squares X2+y2 where X and Y are arbitrary continuous random variables. The techniques employed, which utilize copulas and their properties, show that the bounds are pointwise best-possible when X and Y are symmetric about 0 and yield expressions which can be evaluated explicitly when X and Y have a common distribution function which is concave on (0,oo). Similar results are obtained for the radial error (X2+y2) 1/2. The important case where X and Y are normally distributed is discussed, and here best-possible bounds on the circular probable error are also obtained.

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Best-possible bounds for the distribution of a sum — a problem of Kolmogorov

Cited by 171 publications

References 3 publications

Bounds on the value-at-risk for the sum of possibly dependent risks

Bounds on the value-at-risk for the sum of possibly dependent risks

Possibility transformation of the sum of two symmetric unimodal independent/comonotone random variables

Bounds for distribution functions of sums of squares and radial errors

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