Electronic Journal of Statistics

Partial and average copulas and association measures

Irène Gijbels, Marek Omelka, and Noël Veraverbeke

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Abstract

For a pair $(Y_{1},Y_{2})$ of random variables there exist several measures of association that characterize the dependence between $Y_{1}$ and $Y_{2}$ by means of one single value. Classical examples are Pearson’s correlation coefficient, Kendall’s tau and Spearman’s rho. For the situation where next to the pair $(Y_{1},Y_{2})$ there is also a third variable $X$ present, so-called partial association measures, such as a partial Pearson’s correlation coefficient and a partial Kendall’s tau, have been proposed in the 1940’s. Following criticism on e.g. partial Kendall’s tau, better alternatives to these original partial association measures appeared in the literature: the conditional association measures, e.g. conditional Kendall’s tau, and conditional Spearman’s rho. Both, unconditional and conditional association measures can be expressed in terms of copulas. Even in case the dependence structure between $Y_{1}$ and $Y_{2}$ is influenced by a third variable $X$, we still want to be able to summarize the level of dependence by one single number. In this paper we discuss two different ways to do so, leading to two relatively new concepts: the (new concept of) partial Kendall’s tau, and the average Kendall’s tau. We provide a unifying framework for the diversity of concepts: global (or unconditional) association measures, conditional association measures, and partial and average association measures. The main contribution is that we discuss estimation of the newly-defined concepts: the partial and average copulas and association measures, and establish theoretical results for the estimators. The various concepts of association measures are illustrated on a real data example.

Article information

Source
Electron. J. Statist., Volume 9, Number 2 (2015), 2420-2474.

Dates
Received: October 2014
First available in Project Euclid: 19 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1447943703

Digital Object Identifier
doi:10.1214/15-EJS1077

Mathematical Reviews number (MathSciNet)
MR3425363

Zentralblatt MATH identifier
1327.62208

Subjects
Primary: 62G05: Estimation 62H20: Measures of association (correlation, canonical correlation, etc.)
Secondary: 62G20: Asymptotic properties

Keywords
Average copula conditional copula empirical copula process nonparametric estimation partial copula unconditional copula smoothing weak convergence

Citation

Gijbels, Irène; Omelka, Marek; Veraverbeke, Noël. Partial and average copulas and association measures. Electron. J. Statist. 9 (2015), no. 2, 2420--2474. doi:10.1214/15-EJS1077. https://projecteuclid.org/euclid.ejs/1447943703


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References

  • Acar, E. F., Genest, C. and Nešlehová, J. (2012). Beyond simplified pair-copula constructions., J. Multivariate Anal. 110 74–90.
  • Akritas, M. G. andVan Keilegom, I. (2001). Non-parametric Estimation of the Residual Distribution., Scand. J. Statist. 28 549–567.
  • Bedford, T. and Cooke, R. M. (2002). Vines: a new graphical model for dependent random variables., Ann. Statist. 30 1031–1068.
  • Bergsma, W. P. (2004). Testing conditional independence for continuous random variables., EURANDOM-report.
  • Bergsma, W. P. (2011). Nonparametric testing of conditional independence by means of the partial copula., Arxiv preprint, arXiv:1101.4607.
  • Burda, M. and Prokhorov, A. (2014). Copula based factorization in Bayesian multivariate infinite mixture models., J. Multivariate Anal. 127 200–213.
  • Cabrera, J. L. O. (2012). locpol: Kernel local polynomial regression R package version, 0.6-0.
  • Chen, S. X. and Huang, T. M. (2007). Nonparametric estimation of copula functions for dependence modelling., Canad. J. of Statist. 35 265–282.
  • Cook, R. D. and Johnson, M. E. (1981). A family of distributions for modelling non-elliptically symmetric multivariate data., J. R. Stat. Soc. Ser. B Stat. Methodol. 43 210–218.
  • Cramér, H. (1946)., Mathematical methods of statistics. Princeton University Press.
  • De la Peña, V. H. and Giné, E. (1999)., Decoupling: from dependence to independence. Springer, New York.
  • Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés., Acad. Roy. Belg. Bull. Cl. Sci. 65 274–292.
  • Fermanian, J.-D., Radulovič, D. and Wegkamp, M. (2004). Weak convergence of empirical copula processes., Bernoulli 10 847–860.
  • Gänssler, P. and Stute, W. (1987)., Seminar on Empirical Processes. Birkhäuser, Basel.
  • Gijbels, I. and Mielniczuk, J. (1990). Estimating the density of a copula function., Comm. Statist. Theory Methods 19 445–464.
  • Gijbels, I., Omelka, M. and Veraverbeke, N. (2015). Estimation of a copula when a covariate affects only marginal distributions., Scand. J. Statist. To appear. DOI: 10.1111/sjos.12154.
  • Gijbels, I., Veraverbeke, N. and Omelka, M. (2011). Conditional copulas, association measures and their application., Comput. Stat. Data An. 55 1919–1932.
  • Goodman, L. A. (1959). Partial tests for partial taus., Biometrika 46 425–432.
  • Gripenberg, G. (1992). Confidence intervals for partial rank correlations., J. Amer. Statist. Assoc. 87 546–551.
  • Hobæk Haff, I., Aas, K. and Frigessi, A. (2010). On the simplified pair-copula construction–Simply useful or too simplistic?, J. Multivariate Anal. 101 1296–1310.
  • Joe, H. (2006). Generating random correlation matrices based on partial correlations., J. Multivariate Anal. 97 2177–2189.
  • Kauermann, G. and Schellhase, C. (2014). Flexible pair-copula estimation in D-vines using bivariate penalized splines., Statist. Comput. 24 1081–1100.
  • Kendall, M. G. (1942). Partial Rank Correlation., Biometrika 32 277–283.
  • Kim, J. M., Jung, Y. S., Choi, T. and Sungur, E. A. (2011). Partial correlation with copula modeling., Comput. Stat. Data An. 55 1357–1366.
  • Kojadinovic, I. and Yan, J. (2010). Modeling multivariate distributions with continuous margins using the copula R package., J. Statist. Software 34 1–20.
  • Korn, E. L. (1984). The ranges of limiting values of some partial correlations under conditional independence., Amer. Statist. 38 61–62.
  • Larsson, M., Nešlehová, J. et al. (2011). Extremal behavior of Archimedean copulas., Adv. in Appl. Probab. 43 195–216.
  • Nelsen, R. B. (2006)., An Introduction to Copulas. Springer, New York Second Edition.
  • Nelson, P. I. and Yang, S.-S. (1988). Some properties of Kendall’s partial rank correlation coefficient., Statist. Probab. Lett. 6 147–150.
  • Neumeyer, N. andVan Keilegom, I. (2010). Estimating the error distribution in nonparametric multiple regression with applications to model testing., J. Multivariate Anal. 101 1067–1078.
  • Nolan, D. and Pollard, D. (1987). U-processes: Rates of Convergence., Ann. Statist. 15 780–799.
  • Ojeda, J. (2008). Hölder continuity properties of the local polynomial estimator., Pre-publicaciones del Seminario Matemático “García de Galdeano”. Paper available at http://www.unizar.es/galdeano/preprints/pre08.html.
  • Omelka, M., Gijbels, I. and Veraverbeke, N. (2009). Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing., Ann. Statist. 37 3023–3058. arxiv:0908.4530
  • Omelka, M., Veraverbeke, N. and Gijbels, I. (2013). Bootstrapping the conditional copula., J. Statist. Plann. Inference 143 1–23.
  • Patton, J. A. (2006). Modeling asymmetric exchange rate dependence., Internat. Econom. Rev. 47 527–556.
  • Schmid, F. and Schmidt, R. (2007). Multivariate conditional versions of Spearman’s rho and related measures of tail dependence., J. Multivariate Anal. 98 1123–1140.
  • Segers, J. (2012). Weak convergence of empirical copula processes under nonrestrictive smoothness assumptions., Bernoulli 18 764–782.
  • Serfling, R. J. (1980)., Approximation Theorems of Mathematical Statistics. Wiley, New York.
  • Song, K. (2009). Testing conditional independence via Rosenblatt transforms., Ann. Statist. 37 4011–4045.
  • Tsukuhara, H. (2005). Semiparametric estimation in copula models., Canad. J. of Statist. 33 357–375.
  • van der Vaart, A. W. (2000)., Asymptotic Statistics. Cambridge University Press, New York.
  • van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes. Springer, New York.
  • Veraverbeke, N., Gijbels, I. and Omelka, M. (2014). Pre-adjusted nonparametric estimation of a conditional distribution function., J. R. Stat. Soc. Ser. B Stat. Methodol. 76 399–438.
  • Veraverbeke, N., Omelka, M. and Gijbels, I. (2011). Estimation of a conditional copula and association measures., Scand. J. Statist. 38 766–780.