Abstract
Measuring association, or the lack of it, between variables plays an important role in a variety of research areas, including education,which is of our primary interest in this paper. Given, for example, student marks on several study subjects, we may for a number of reasons be interested in measuring the lack of comonotonicity (LOC) between the marks, which rarely follow monotone, let alone linear, patterns. For this purpose, in this paperwe explore a novel approach based on a LOCindex,which is related to, yet substantially different from, Eckhard Liebscher’s recently suggested coefficient of monotonically increasing dependence. To illustrate the new technique,we analyze a data-set of student marks on mathematics, reading and spelling.
References
[1] Bebbington, M., Lai, C. D. and Zitikis, R. (2011). Modelling deceleration in senescent mortality. Math. Popul. Stud. 18(1), 18–37. 10.1080/08898480.2011.540173Search in Google Scholar
[2] Brazauskas, V., Jones, B. L. and Zitikis, R. (2015). Trends in disguise. Ann. Actuar. Sci. 9(1), 58–71. 10.1017/S1748499514000232Search in Google Scholar
[3] Carlier, G. and Dana, R. A. (2003). Pareto efficient insurance contracts when the insurer’s cost function is discontinuous. Econom. Theory 21(4), 871–893. 10.1007/s00199-002-0281-zSearch in Google Scholar
[4] Carlier, G. and Dana, R. (2005). Rearrangement inequalities in non-convex insurance models. J.Math. Econom. 41(4-5), 485– 503. 10.1016/j.jmateco.2004.12.004Search in Google Scholar
[5] Carlier, G. and Dana, R. (2008). Two-persons efficient risk-sharing and equilibria for concave law-invariant utilities. Econom. Theory 36(2), 189–223. 10.1007/s00199-007-0266-zSearch in Google Scholar
[6] Carlier, G. and Dana, R. (2011). Optimal demand for contingent claims when agents have lawinvariant utilities.Math. Finance 21(2), 169–201. Search in Google Scholar
[7] Chernozhukov, V., Fernandez-Val, I. and Galichon, A. (2009). Improving point and interval estimators of monotone function by rearrangement. Biometrika 96, 559–575. 10.1093/biomet/asp030Search in Google Scholar
[8] Chernozhukov, V., Fernandez-Val, I. and Galichon, A. (2010). Quantile and probability curveswithout crossing. Econometrica 78(3), 1093–1125. 10.3982/ECTA7880Search in Google Scholar
[9] Dana, R. and Scarsini, M. (2007). Optimal risk sharing with background risk. J. Econom. Theory 133(1), 152–176. 10.1016/j.jet.2005.10.002Search in Google Scholar
[10] Denneberg, D. (1994). Non-additive Measure and Integral. Kluwer Academic Publishers Group, Dordrecht. 10.1007/978-94-017-2434-0Search in Google Scholar
[11] Dhaene, J., Linders, D., Schoutens,W. and Vyncke, D. (2012). The herd behavior index: a new measure for the implied degree of co-movement in stock markets. Insurance Math. Econom. 50(3), 357–370. 10.1016/j.insmatheco.2012.01.005Search in Google Scholar
[12] Dhaene J., Linders D., Schoutens W. and Vyncke D. (2014). A multivariate dependence measure for aggregating risks. J. Comput. Appl. Math. 263(1), 78–87. 10.1016/j.cam.2013.12.010Search in Google Scholar
[13] Dhaene, J., Vanduffel, S., Goovaerts, M. J., Kaas, R., Tang, Q. and Vyncke, D. (2006). Risk measures and comonotonicity: a review. Stoch. Models 22(4), 573–606. 10.1080/15326340600878016Search in Google Scholar
[14] Fan, J. (1992). Design-adaptive nonparametric regression. J. Amer. Statist. Assoc. 87(420), 998–1004. 10.1080/01621459.1992.10476255Search in Google Scholar
[15] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman & Hall, London. Search in Google Scholar
[16] Fan, J. and Gijbels, I. (2000). Local polynomial fitting. In: Schimek, M.G. (ed.), Smoothing and Regression: Approaches, Computation, and Application. Wiley, New York, 229–276. 10.1002/9781118150658.ch9Search in Google Scholar
[17] Furman, E. and Zitikis, R. (2008). Weighted risk capital allocations. Insurance Math. Econom. 43(2), 263–269. 10.1016/j.insmatheco.2008.07.003Search in Google Scholar
[18] Gavrilov, L. A. and Gavrilova, N. S. (1991). The Biology of Life Span: A Quantitative Approach. Harwood Academic Publishers, New York. Search in Google Scholar
[19] Hardy, G. H., Littlewood, J.E. and Pólya, G. (1988). Inequalities (2nd edition). Cambridge University Press, Cambridge. Search in Google Scholar
[20] He, X. and Zhou, X. (2011). Portfolio choice via quantile. Math. Finance 21(2), 203–231. Search in Google Scholar
[21] Jaworski, P., Durante, F. and Härdle, W. (2013). Copulae in Mathematical and Quantitative Finance. Springer, Berlin. 10.1007/978-3-642-35407-6Search in Google Scholar
[22] Jaworski, P., Durante, F., Härdle, W. and Rychlik, T. (2010). Copula Theory and Its Applications. Springer, Heidelberg. 10.1007/978-3-642-12465-5Search in Google Scholar
[23] Jin, H. and Zhou, X. (2008). Behavioral portfolio selection in continuous time. Math. Finance 18(3), 385–426. 10.1111/j.1467-9965.2008.00339.xSearch in Google Scholar
[24] Koch, I. and De Schepper, A. (2011). Measuring comonotonicity in m-dimensional vectors. ASTIN Bull. 41(1), 191–213. Search in Google Scholar
[25] Koenker, R. (2005). Quantile Regression. Cambridge University Press, Cambridge. 10.1017/CBO9780511754098Search in Google Scholar
[26] Koenker, R. (2015). quantreg: Quantile Regression. R package version 5.11, URL http://cran.rproject. org/web/packages/quantreg/index.html Search in Google Scholar
[27] Lai, C. D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York. Search in Google Scholar
[28] Lehmann, E. L. (1966). Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153. 10.1214/aoms/1177699260Search in Google Scholar
[29] Li, H. and Li, X. (2013). Stochastic Orders in Reliability and Risk: in Honor of Professor Moshe Shaked. Springer, New York. Search in Google Scholar
[30] Liebscher, E. (2014). Copula-based dependence measures. Depend. Model. 2(1), 49–64. 10.2478/demo-2014-0004Search in Google Scholar
[31] Nelsen, R. B. (2006). An Introduction to Copulas (2nd Edition). Springer, New York. Search in Google Scholar
[32] Panik, M. J. (2014). Growth Curve Modeling: Theory and Applications. John Wiley & Sons, New York. 10.1002/9781118763971Search in Google Scholar
[33] Pearl, J. (2009). Causality: Models, Reasoning, and Inference (2nd Edition). Cambridge University Press, New York. Search in Google Scholar
[34] Qoyyimi, D. T. and Zitikis, R. (2014). Measuring the lack of monotonicity in functions. Math. Sci. 39(2), 107–117. 10.2139/ssrn.2413232Search in Google Scholar
[35] Quiggin, J. (1982). A theory of anticipated utility. J. Econom. Behav. Organ. 3(4), 323–343. 10.1016/0167-2681(82)90008-7Search in Google Scholar
[36] Quiggin, J. (2012). Generalized Expected Utility Theory: The Rank Dependent Model. Springer Science & Business Media. Search in Google Scholar
[37] R Core Team (2013). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/. Search in Google Scholar
[38] Ruppert, D., Sheather, S. J. and Wand, M. P. (1995). An effective bandwidth selector for local least squares regression. J. Amer. Statist. Assoc. 90(432), 1257–1270. 10.1080/01621459.1995.10476630Search in Google Scholar
[39] Ruppert, D. and Wand, M. P. (1994). Multivariate locally weighted least squares regression. Ann. Statist. 22(3), 1346–1370. 10.1214/aos/1176325632Search in Google Scholar
[40] Rüschendorf, L. (1983). Solution of a statistical optimization problem by rearrangement methods. Metrika 30(1), 55–61. 10.1007/BF02056901Search in Google Scholar
[41] Scarsini, M. (1984). On measures of concordance. Stochastica 8(3), 201–218. Search in Google Scholar
[42] Schweizer, B. and Wolff, E. F. (1981). On nonparametric measures of dependence for random variables. Ann. Statist. 9(4), 879–885. 10.1214/aos/1176345528Search in Google Scholar
[43] Seber, G. A. F. and Wild, C. J. (1989). Nonlinear Regression. John Wiley & Sons, New York. 10.1002/0471725315Search in Google Scholar
[44] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York. 10.1007/978-0-387-34675-5Search in Google Scholar
[45] Thorndike, R. M. and Thorndike-Christ, T. (2010). Measurement and Evaluation in Psychology and Education (8th Edition). Pearson Prentice Hall. Search in Google Scholar
[46] Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman & Hall, London. 10.1007/978-1-4899-4493-1Search in Google Scholar
[47] Wand, M. P. and Ripley, B. (2014). KernSmooth: Functions for Kernel Smoothing. R package version 2.23-13, URL http://CRAN.R-project.org/package=KernSmooth. Search in Google Scholar
[48] Yu, K. and Jones, M. C. (1997). A comparison of local constant and local linear regression quantile estimator. Comput. Statist. Data Anal. 25(2), 159–166. 10.1016/S0167-9473(97)00006-6Search in Google Scholar
© 2015 Danang Teguh Qoyyimi and Ricardas Zitikis
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