Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
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
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.
Opis fizyczny
  • [1] Balakrishnan N., Lai C.-D., Continuous Bivariate Distributions, 2nd ed., Springer, New York, 2009
  • [2] Broll U., Egozcue M., Wong W.-K., Zitikis R., Prospect theory, indifference curves, and hedging risks, Appl. Math. Res. Express. AMRX, 2010, 2, 142–153
  • [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
  • [5] Denuit M., Dhaene J., Goovaerts M., Kaas R., Actuarial Theory for Dependent Risks: Measures, Orders and Models, John Wiley & Sons, Chichester, 2005
  • [6] Dudley D.M., Norvaiša R., Differentiability of Six Operators on Nonsmooth Functions and p-Variation, Lecture Notes in Math., 1703, Springer, New York, 1999
  • [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
  • [11] Furman E., Zitikis R., General Stein-type covariance decompositions with applications to insurance and finance, Astin Bull., 2010, 40(1), 369–375
  • [12] Kowalczyk T., Pleszczynska E., Monotonic dependence functions of bivariate distributions, Ann. Statist., 1977, 5(6), 1221–1227
  • [13] Lehmann E.L., Some concepts of dependence, Ann. Math. Statist., 1966, 37(5), 1137–1153
  • [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
  • [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
  • [19] Wright R., Expectation dependence of random variables, with an application in portfolio theory, Theory and Decision, 1987, 22(2), 111–124
  • [20] Zitikis R., Grüss’s inequality, its probabilistics interpretation, and a sharper bound, J. Math. Inequal., 2009, 3(1), 15–20
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.