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.
@article{bwmeta1.element.doi-10_2478_s11533-011-0088-x,
author = {Mart\'\i n Egozcue and Luis Garc\'\i a and Wing-Keung Wong and Ri\v cardas Zitikis},
title = {Gr\"uss-type bounds for covariances and the notion of quadrant dependence in expectation},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {1288-1297},
zbl = {1228.60028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0088-x}
}
Martín Egozcue; Luis García; Wing-Keung Wong; Ričardas Zitikis. Grüss-type bounds for covariances and the notion of quadrant dependence in expectation. Open Mathematics, Tome 9 (2011) pp. 1288-1297. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0088-x/
[1] Balakrishnan N., Lai C.-D., Continuous Bivariate Distributions, 2nd ed., Springer, New York, 2009 | Zbl 1267.62028
[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 | Zbl 1231.91097
[3] Cerone P., Dragomir S.S., Mathematical Inequalities, CRC Press, Boca Raton, 2011 | Zbl 1298.26006
[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
[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 | Zbl 1218.46003
[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 | Zbl 05636867
[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 | Zbl 1200.62069
[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 | Zbl 1189.62163
[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 | Zbl 1191.62097
[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 | Zbl 0374.62051
[13] Lehmann E.L., Some concepts of dependence, Ann. Math. Statist., 1966, 37(5), 1137–1153 http://dx.doi.org/10.1214/aoms/1177699260 | Zbl 0146.40601
[14] Matuła P., On some inequalities for positively and negatively dependent random variables with applications, Publ. Math. Debrecen, 2003, 63(4), 511–522 | Zbl 1048.60016
[15] Matuła P., A note on some inequalities for certain classes of positively dependent random variables, Probab. Math. Statist., 2004, 24(1), 17–26 | Zbl 1061.60013
[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 | Zbl 1217.60017
[17] McNeil A.J., Frey R., Embrechts P., Quantitative Risk Management, Princet. Ser. Finance, Princeton University Press, Princeton, 2005 | Zbl 1089.91037
[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 | Zbl 1207.60012
[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 | Zbl 0607.90007
[20] Zitikis R., Grüss’s inequality, its probabilistics interpretation, and a sharper bound, J. Math. Inequal., 2009, 3(1), 15–20 | Zbl 1175.26060