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2011 | 9 | 6 | 1288-1297
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
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.
Wydawca
Czasopismo
Rocznik
Tom
9
Numer
6
Strony
1288-1297
Opis fizyczny
Daty
wydano
2011-12-01
online
2011-09-23
Twórcy
autor
Bibliografia
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  • [3] Cerone P., Dragomir S.S., Mathematical Inequalities, CRC Press, Boca Raton, 2011
  • [4] Cuadras C.M., On the covariance between functions, J. Multivariate Anal., 2002, 81(1), 19–27 http://dx.doi.org/10.1006/jmva.2001.2000
  • [5] Denuit M., Dhaene J., Goovaerts M., Kaas R., Actuarial Theory for Dependent Risks: Measures, Orders and Models, John Wiley & Sons, Chichester, 2005 http://dx.doi.org/10.1002/0470016450
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  • [7] Dudley D.M., Norvaiša R., Concrete Functional Calculus, Springer Monogr. Math., Springer, New York, 2011
  • [8] Egozcue M., Fuentes Garcia L., Wong W.-K., On some covariance inequalities for monotonic and non-monotonic functions, JIPAM. J. Inequal. Pure Appl. Math., 2009, 10(3), #75
  • [9] Egozcue M., Fuentes García L., Wong W.-K., Zitikis R., Grüss-type bounds for the covariance of transformed random variables, J. Inequal. Appl., 2010, ID 619423
  • [10] Furman E., Zitikis R., Weighted risk capital allocations, Insurance Math. Econom., 2008, 43(2), 263–269 http://dx.doi.org/10.1016/j.insmatheco.2008.07.003
  • [11] Furman E., Zitikis R., General Stein-type covariance decompositions with applications to insurance and finance, Astin Bull., 2010, 40(1), 369–375 http://dx.doi.org/10.2143/AST.40.1.2049234
  • [12] Kowalczyk T., Pleszczynska E., Monotonic dependence functions of bivariate distributions, Ann. Statist., 1977, 5(6), 1221–1227 http://dx.doi.org/10.1214/aos/1176344006
  • [13] Lehmann E.L., Some concepts of dependence, Ann. Math. Statist., 1966, 37(5), 1137–1153 http://dx.doi.org/10.1214/aoms/1177699260
  • [14] Matuła P., On some inequalities for positively and negatively dependent random variables with applications, Publ. Math. Debrecen, 2003, 63(4), 511–522
  • [15] Matuła P., A note on some inequalities for certain classes of positively dependent random variables, Probab. Math. Statist., 2004, 24(1), 17–26
  • [16] Matuła P., Ziemba M., Generalized covariance inequalities. Cent. Eur. J. Math., 2011, 9(2), 281–293 http://dx.doi.org/10.2478/s11533-011-0006-2
  • [17] McNeil A.J., Frey R., Embrechts P., Quantitative Risk Management, Princet. Ser. Finance, Princeton University Press, Princeton, 2005
  • [18] Niezgoda M., New bounds for moments of continuous random variables, Comput. Math. Appl., 2010, 60(12), 3130–3138 http://dx.doi.org/10.1016/j.camwa.2010.10.018
  • [19] Wright R., Expectation dependence of random variables, with an application in portfolio theory, Theory and Decision, 1987, 22(2), 111–124 http://dx.doi.org/10.1007/BF00126386
  • [20] Zitikis R., Grüss’s inequality, its probabilistics interpretation, and a sharper bound, J. Math. Inequal., 2009, 3(1), 15–20
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0088-x
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