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BY 4.0 license Open Access Published by De Gruyter Open Access July 27, 2020

Bivariate box plots based on quantile regression curves

  • Jorge Navarro EMAIL logo
From the journal Dependence Modeling


In this paper, we propose a procedure to build bivariate box plots (BBP). We first obtain the theoretical BBP for a random vector (X, Y). They are based on the univariate box plot of X and the conditional quantile curves of Y|X. They can be computed from the copula of (X, Y) and the marginal distributions. The main advantage of these BBP is that the coverage probabilities of the regions are distribution-free. So they can be selected by the users with the desired probabilities and they can be used to perform fit tests. Three reasonable options are proposed. They are illustrated with two examples from a normal model and an exponential model with a Clayton copula. Moreover, several methods to estimate the theoretical BBP are discussed. The main ones are based on linear and non-linear quantile regression. The others are based on empirical estimators and parametric and non-parametric (kernel) copula estimations. All of them can be used to get empirical BBP. Some extensions for the multivariate case are proposed as well.

MSC 2010: 62G99; 62G07


[1] Arnold, B. C., E. Castillo, and J. M. Sarabia (1999). Conditional Specification of Statistical Models. Springer, New York.Search in Google Scholar

[2] Bernard, C. and C. Czado (2015). Conditional quantiles and tail dependence. J. Multivariate Anal. 138, 104–126.10.1016/j.jmva.2015.01.011Search in Google Scholar

[3] Bernardi, M., F. Durante, and P. Jaworski (2017). CoVaR of families of copulas. Statist. Probab. Lett. 120, 8–17.10.1016/j.spl.2016.09.005Search in Google Scholar

[4] Brownlee, K. A. (1960). Statistical Theory and Methodology in Science and Engineering. John Wiley & Sons, New York.Search in Google Scholar

[5] Chen, X. (2011). A new generalization of Chebyshev inequality for random vectors. Available at in Google Scholar

[6] Fernández-Ponce, J. M. and A. Suárez-Lloréns (2002). Central regions for bivariate distributions. Austrian J. Stat. 31(2&3), 141–156.Search in Google Scholar

[7] Greenwood, P. E. and M. S. Nikulin (1996). A Guide to Chi-Squared Testing. John Wiley & Sons, New York.Search in Google Scholar

[8] Hyndman, R. J. and Y. Fan (1996). Sample quantiles in statistical packages. Amer. Statist. 50(4), 361–365.Search in Google Scholar

[9] Jaworski, P. (2017). On conditional value at risk (CoVaR) for tail-dependent copulas. Depend. Model. 5, 1–19.10.1515/demo-2017-0001Search in Google Scholar

[10] Koenker, R. (2005). Quantile Regression. Cambridge University Press.10.1017/CBO9780511754098Search in Google Scholar

[11] Koenker, R. and G. Bassett Jr. (1978). Regression quantiles. Econometrica 46(1), 33–50.10.2307/1913643Search in Google Scholar

[12] Koenker, R. and B. J. Park (1996). An interior point algorithm for nonlinear quantile regression. J. Econometrics 71(1-2), 265–283.10.1016/0304-4076(96)84507-6Search in Google Scholar

[13] Koenker, R., S. Portnoy, P. Tian, A. Zeileis, P. Grosjean, C. Moler, and B. D. Ripley (2020). Quantreg: Quantile Regression. R package version 5.55. Available on CRAN.Search in Google Scholar

[14] Mardia, K. V., J. T. Kent, and J. M. Bibby (1979). Multivariate Analysis. Academic Press, London.Search in Google Scholar

[15] Nagler, T. (2014). Kernel Methods for Vine Copula Estimation. Master’s thesis, Technische Universität München, Germany.Search in Google Scholar

[16] Navarro, J. (2014). A note on confidence regions based on the bivariate Chebyshev inequality. Istatistik 7(1), 1–14.Search in Google Scholar

[17] Navarro, J. (2016). A very simple proof of the multivariate Chebyshev’s inequality. Comm. Statist. Theory Methods 45(12), 3458–3463.10.1080/03610926.2013.873135Search in Google Scholar

[18] Navarro, J. and M. A. Sordo (2018). Stochastic comparisons and bounds for conditional distributions by using copula properties. Depend. Model. 6, 156–177.10.1515/demo-2018-0010Search in Google Scholar

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

[20] Rousseeuw, P. J., I. Ruts, and J. W. Tukey (1999). The bagplot: A bivariate boxplot. Amer. Statist. 53(4), 382–387.Search in Google Scholar

[21] Sumarjaya, I. W. (2017). A survey of kernel-type estimators for copula and their applications. J. Phys.: Conf. Ser. 893, Article ID 012027, 6 pages.Search in Google Scholar

Received: 2020-04-21
Accepted: 2020-06-20
Published Online: 2020-07-27

© 2020 Jorge Navarro, published by De Gruyter

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

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