Abstract
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.
[1] Balakrishnan N., Lai C.-D., Continuous Bivariate Distributions, 2nd ed., Springer, New York, 2009 10.1007/b101765_6Search in Google Scholar
[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 10.1093/amrx/abq013Search in Google Scholar
[3] Cerone P., Dragomir S.S., Mathematical Inequalities, CRC Press, Boca Raton, 2011 10.1201/b10483Search in Google Scholar
[4] Cuadras C.M., On the covariance between functions, J. Multivariate Anal., 2002, 81(1), 19–27 http://dx.doi.org/10.1006/jmva.2001.200010.1006/jmva.2001.2000Search in Google Scholar
[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/10.1002/047001645010.1002/0470016450Search in Google Scholar
[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 10.1007/BFb0100744Search in Google Scholar
[7] Dudley D.M., Norvaiša R., Concrete Functional Calculus, Springer Monogr. Math., Springer, New York, 2011 10.1007/978-1-4419-6950-7Search in Google Scholar
[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 Search in Google Scholar
[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 10.1155/2010/619423Search in Google Scholar
[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.2008.07.00310.1016/j.insmatheco.2008.07.003Search in Google Scholar
[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.40.1.204923410.2143/AST.40.1.2049234Search in Google Scholar
[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/117634400610.1214/aos/1176344006Search in Google Scholar
[13] Lehmann E.L., Some concepts of dependence, Ann. Math. Statist., 1966, 37(5), 1137–1153 http://dx.doi.org/10.1214/aoms/117769926010.1214/aoms/1177699260Search in Google Scholar
[14] Matuła P., On some inequalities for positively and negatively dependent random variables with applications, Publ. Math. Debrecen, 2003, 63(4), 511–522 Search in Google Scholar
[15] Matuła P., A note on some inequalities for certain classes of positively dependent random variables, Probab. Math. Statist., 2004, 24(1), 17–26 Search in Google Scholar
[16] Matuła P., Ziemba M., Generalized covariance inequalities. Cent. Eur. J. Math., 2011, 9(2), 281–293 http://dx.doi.org/10.2478/s11533-011-0006-210.2478/s11533-011-0006-2Search in Google Scholar
[17] McNeil A.J., Frey R., Embrechts P., Quantitative Risk Management, Princet. Ser. Finance, Princeton University Press, Princeton, 2005 Search in Google Scholar
[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.2010.10.01810.1016/j.camwa.2010.10.018Search in Google Scholar
[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/BF0012638610.1007/BF00126386Search in Google Scholar
[20] Zitikis R., Grüss’s inequality, its probabilistics interpretation, and a sharper bound, J. Math. Inequal., 2009, 3(1), 15–20 10.7153/jmi-03-02Search in Google Scholar
© 2011 Versita Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.