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BY 4.0 license Open Access Published by De Gruyter Open Access May 11, 2019

On the lower bound of Spearman’s footrule

  • Sebastian Fuchs EMAIL logo and Yann McCord
From the journal Dependence Modeling

Abstract

Úbeda-Flores showed that the range of multivariate Spearman’s footrule for copulas of dimension d ≥ 2 is contained in the interval [−1/d, 1], that the upper bound is attained exclusively by the upper Fréchet-Hoeffding bound, and that the lower bound is sharp in the case where d = 2. The present paper provides characterizations of the copulas attaining the lower bound of multivariate Spearman’s footrule in terms of the copula measure but also via the copula’s diagonal section.

MSC 2010: 62H05; 62H20

References

[1] Diaconis, P. and R. L. Graham (1977). Spearman’s footrule as a measure of disarray. J. R. Stat. Soc. Ser. B. Stat. Methodol. 39(2), 262–268.10.1111/j.2517-6161.1977.tb01624.xSearch in Google Scholar

[2] Dolati, A. and M. Úbeda-Flores (2006). On measures of multivariate concordance. J. Probab. Stat. Sci. 4(2), 147–163.Search in Google Scholar

[3] Durante, F. and J. Fernández-Sánchez (2010). Multivariate shuffles and approximation of copulas. Statist. Probab. Lett. 80(22-23), 1827–1834.10.1016/j.spl.2010.08.008Search in Google Scholar

[4] Durante, F. and C. Sempi (2016). Principles of Copula Theory. CRC Press, Boca Raton FL.Search in Google Scholar

[5] Fernández-Sánchez, J. and W. Trutschnig (2015). Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and iterated function systems. J. Theoret. Probab. 28(4), 1311–1336.10.1007/s10959-014-0541-4Search in Google Scholar

[6] Fuchs, S. (2014). Multivariate copulas: Transformations, symmetry, order and measures of concordance. Kybernetika 50(5), 725–743.10.14736/kyb-2014-5-0725Search in Google Scholar

[7] Fuchs, S. (2016a). A biconvex form for copulas. Depend. Model. 4, 63–75.10.1515/demo-2016-0003Search in Google Scholar

[8] Fuchs, S. (2016b). Copula-induced measures of concordance. Depend. Model. 4, 205–214.10.1515/demo-2016-0011Search in Google Scholar

[9] Fuchs, S., Y. McCord, and K. D. Schmidt (2018). Characterizations of copulas attaining the bounds of multivariate Kendall’s tau. J. Optim. Theory Appl. 178(2), 424–438.10.1007/s10957-018-1285-6Search in Google Scholar

[10] Fuchs, S. and K. D. Schmidt (2014). Bivariate copulas: Transformations, asymmetry and measures of concordance. Kybernetika 50(1), 109–125.10.14736/kyb-2014-1-0109Search in Google Scholar

[11] Genest, C., J. Nešlehová, and N. Ben Ghorbal (2010). Spearman’s footrule and Gini’s gamma: a review with complements. J. Nonparametr. Stat. 22(8), 937–954.10.1080/10485250903499667Search in Google Scholar

[12] Kim, B. S., S. Y. Rha, G. B. Cho, and H. C. Chung (2004). Spearman’s footrule as ameasure of cDNA microarray reproducibility. Genomics 84(2), 441–448.10.1016/j.ygeno.2004.02.015Search in Google Scholar PubMed

[13] Lee, P. H. and P. L. H. Yu (2010). Distance-based tree models for ranking data. Comput. Statist. Data Anal. 54(6), 1672–1682.10.1016/j.csda.2010.01.027Search in Google Scholar

[14] Nelsen, R. B. (2006). An Introduction to Copulas. Second edition. Springer, New York.Search in Google Scholar

[15] Pérez, A. and M. Prieto-Alaiz (2016). Measuring the dependence among dimensions of welfare: A study based on Spearman’s footrule and Gini’s gamma. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 24(1), 87–105.10.1142/S0218488516400055Search in Google Scholar

[16] Sen, P. K., I. A. Salama, and D. Quade (2011). Spearman’s footrule: Asymptotics in applications. Chil. J. Stat. 2(1), 3–20.Search in Google Scholar

[17] Taylor, M. D. (2016). Multivariate measures of concordance for copulas and their marginals. Depend. Model. 4, 224–236.10.1515/demo-2016-0013Search in Google Scholar

[18] Úbeda-Flores, M. (2005). Multivariate versions of Blomqvist’s beta and Spearman’s footrule. Ann. Inst. Statist.Math. 57(4), 781–788.10.1007/BF02915438Search in Google Scholar

Received: 2018-11-20
Accepted: 2019-04-07
Published Online: 2019-05-11

© 2019 Sebastian Fuchs et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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