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1
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Grüss-type bounds for covariances and the notion of quadrant dependence in expectation

100%
Egozcue M., García L., Wong W.-K., Zitikis R.
Open Mathematics
|
2011
|
tom 9
|
nr 6
1288-1297
EN
We show that Grüss-type probabilistic inequalities for covariances can be considerably sharpened when the underlying random variables are quadrant dependent in expectation (QDE). The herein established covariance bounds not only sharpen the classical Grüss inequality but also improve upon recently derived Grüss-type bounds under the assumption of quadrant dependency (QD), which is stronger than QDE. We illustrate our general results with examples based on specially devised bivariate distributions that are QDE but not QD. Such results play important roles in decision making under uncertainty, and particularly in areas such as economics, finance, and insurance.
2
Content available remote

Quantile of a Mixture with Application to Model Risk Assessment

100%
Bernard C., Vanduffel S.
Dependence Modeling
|
2015
|
tom 3
|
nr 1
EN
We provide an explicit expression for the quantile of a mixture of two random variables. The result is useful for finding bounds on the Value-at-Risk of risky portfolios when only partial dependence information is available. This paper complements the work of [4].
3
Content available remote

The 123 theorem of Probability Theory and Copositive Matrices

76%
Kovačec A., Moreira M. M. R., Martins D. P.
Special Matrices
|
2014
|
tom 2
|
nr 1
EN
Alon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these using copositive matrices instead. By the same method we also give estimates for the real valued case, involving X + Y and X − Y, due to Siegmund-Schultze and von Weizsäcker as generalized by Dong, Li and Li. Furthermore, we formulate a version of the above inequalities as an integral inequality for monotone functions.
4
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Are law-invariant risk functions concave on distributions?

76%
Acciaio B., Svindland G.
Dependence Modeling
|
2013
|
tom 1
54-64
EN
While it is reasonable to assume that convex combinations on the level of random variables lead to a reduction of risk (diversification effect), this is no more true on the level of distributions. In the latter case, taking convex combinations corresponds to adding a risk factor. Hence, whereas asking for convexity of risk functions defined on random variables makes sense, convexity is not a good property to require on risk functions defined on distributions. In this paper we study the interplay between convexity of law-invariant risk functions on random variables and convexity/concavity of their counterparts on distributions. We show that, given a law-invariant convex risk measure, on the level of distributions, if at all, concavity holds true. In particular, this is always the case under the additional assumption of comonotonicity.
5
Content available remote

Seven Proofs for the Subadditivity of Expected Shortfall

76%
Embrechts P., Wang R.
Dependence Modeling
|
2015
|
tom 3
|
nr 1
EN
Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof.With different proofs, different important properties of ES are revealed, such as its dual representation, optimization properties, continuity, consistency with convex order, and natural estimators.
6
Content available remote

Generalized covariance inequalities

64%
Matuła P., Ziemba M.
Open Mathematics
|
2011
|
tom 9
|
nr 2
281-293
EN
We prove some inequalities for the difference between a joint distribution and the product of its marginals for arbitrary absolutely continuous random variables. Some applications of the obtained inequalities are also presented.
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