On the covariance of the asymptotic empirical copula process. (English) Zbl 1211.62030
The authors compare the asymptotic covariance of the empirical process to that of the empirical copula process, where the marginals are estimated by the empirical marginals. The main result of the paper states the somewhat counterintuitive result that under the assumption of a left tail dependent copula the covariance of the empirical copula process is pointwise smaller than the covariance of the copula itself. The authors state some implications of this result for inference of copula based dependence parameters and also for the \(d\)-dimensional case.
Reviewer: Ludger Rüschendorf (Freiburg i. Br.)
MSC:
| 62E20 | Asymptotic distribution theory in statistics |
| 60F17 | Functional limit theorems; invariance principles |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 62G30 | Order statistics; empirical distribution functions |
Keywords:
asymptotic variance; copula; dependence parameter; empirical process; independence; left-tail decreasing; rank-based inferenceReferences:
| [1] | Joe, H., Multivariate Models and Dependence Concepts (1997), Chapman & Hall: Chapman & Hall London · Zbl 0990.62517 |
| [2] | Nelsen, R. B., An Introduction to Copulas (2006), Springer: Springer Berlin · Zbl 1152.62030 |
| [3] | Shorack, G. R.; Wellner, J. A., Empirical Processes with Applications to Statistics (1986), Wiley: Wiley New York · Zbl 1170.62365 |
| [4] | Deheuvels, P., La fonction de dépendance empirique et ses propriétés: Un test non paramétrique d’indépendance, Acad. Roy. Belg. Bull. Cl. Sci., 65, 5, 274-292 (1979) · Zbl 0422.62037 |
| [5] | Berg, D., Copula goodness-of-fit testing: an overview and power comparison, Europ. J. Finance, 15, 675-701 (2009) |
| [6] | Genest, C.; Rémillard, B.; Beaudoin, D., Goodness-of-fit tests for copulas: a review and a power study, Insurance: Math. Econom., 44, 199-213 (2009) · Zbl 1161.91416 |
| [7] | Rüschendorf, L., Asymptotic distributions of multivariate rank order statistics, Ann. Statist., 4, 912-923 (1976) · Zbl 0359.62040 |
| [8] | Fermanian, J.-D.; Radulovic, D.; Wegkamp, M., Weak convergence of empirical copula processes, Bernoulli, 10, 847-860 (2004) · Zbl 1068.62059 |
| [9] | Esary, J. D.; Proschan, F., Relationships among some concepts of bivariate dependence, Ann. Math. Statist., 43, 651-655 (1972) · Zbl 0263.62011 |
| [10] | Lehmann, E. L., Some concepts of dependence, Ann. Math. Statist., 37, 1137-1153 (1966) · Zbl 0146.40601 |
| [11] | Charpentier, A.; Fermanian, J.-D.; Scaillet, O., The estimation of copulas: theory and practice, (Rank, J., Copulas: From Theory to Application in Finance (2007), Risk Publications: Risk Publications London), 35-60 |
| [12] | Genest, C.; Segers, J., Rank-based inference for bivariate extreme-value copulas, Ann. Statist., 37, 2990-3022 (2009) · Zbl 1173.62013 |
| [13] | Garralda-Guillem, A. I., Structure de dépendance des lois de valeurs extrêmes bivariées, C. R. Acad. Sci. Paris Sér. I Math., 330, 593-596 (2000) · Zbl 0951.60014 |
| [14] | Schmid, F.; Schmidt, R., Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence, Metrika, 66, 323-354 (2007) · Zbl 1433.62151 |
| [15] | Scarsini, M., On measures of concordance, Stochastica, 8, 201-218 (1984) · Zbl 0582.62047 |
| [16] | Genest, C.; Nešlehová, J., A primer on copulas for count data, Astin Bull., 38, 475-515 (2007) · Zbl 1274.62398 |
| [17] | Henmi, M., A paradoxical effect of nuisance parameters on efficiency of estimators, J. Japan Statist. Soc., 34, 75-86 (2005) · Zbl 1061.62053 |
| [18] | Denuit, M.; Scaillet, O., Nonparametric tests for positive quadrant dependence, J. Financ. Econ., 2, 422-450 (2004) |
| [19] | van der Vaart, A. W., Asymptotic Statistics (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0910.62001 |
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