## Bernoulli

• Bernoulli
• Volume 23, Number 3 (2017), 1848-1873.

### Probit transformation for nonparametric kernel estimation of the copula density

#### Abstract

Copula modeling has become ubiquitous in modern statistics. Here, the problem of nonparametrically estimating a copula density is addressed. Arguably the most popular nonparametric density estimator, the kernel estimator is not suitable for the unit-square-supported copula densities, mainly because it is heavily affected by boundary bias issues. In addition, most common copulas admit unbounded densities, and kernel methods are not consistent in that case. In this paper, a kernel-type copula density estimator is proposed. It is based on the idea of transforming the uniform marginals of the copula density into normal distributions via the probit function, estimating the density in the transformed domain, which can be accomplished without boundary problems, and obtaining an estimate of the copula density through back-transformation. Although natural, a raw application of this procedure was, however, seen not to perform very well in the earlier literature. Here, it is shown that, if combined with local likelihood density estimation methods, the idea yields very good and easy to implement estimators, fixing boundary issues in a natural way and able to cope with unbounded copula densities. The asymptotic properties of the suggested estimators are derived, and a practical way of selecting the crucially important smoothing parameters is devised. Finally, extensive simulation studies and a real data analysis evidence their excellent performance compared to their main competitors.

#### Article information

Source
Bernoulli, Volume 23, Number 3 (2017), 1848-1873.

Dates
Revised: August 2015
First available in Project Euclid: 17 March 2017

https://projecteuclid.org/euclid.bj/1489737627

Digital Object Identifier
doi:10.3150/15-BEJ798

Mathematical Reviews number (MathSciNet)
MR3624880

Zentralblatt MATH identifier
06714321

#### Citation

Geenens, Gery; Charpentier, Arthur; Paindaveine, Davy. Probit transformation for nonparametric kernel estimation of the copula density. Bernoulli 23 (2017), no. 3, 1848--1873. doi:10.3150/15-BEJ798. https://projecteuclid.org/euclid.bj/1489737627

#### References

• [1] Autin, F., Le Pennec, E. and Tribouley, K. (2010). Thresholding methods to estimate copula density. J. Multivariate Anal. 101 200–222.
• [2] Behnen, K., Hušková, M. and Neuhaus, G. (1985). Rank estimators of scores for testing independence. Statist. Decisions 3 239–262.
• [3] Blumentritt, T. (2011). On copula density estimation and measures of multivariate association. Ph.D. Dissertation, Universität zu Köln.
• [4] Bouezmarni, T., El Ghouch, A. and Taamouti, A. (2013). Bernstein estimator for unbounded copula densities. Stat. Risk Model. 30 343–360.
• [5] Bouezmarni, T., Rombouts, J.V.K. and Taamouti, A. (2010). Asymptotic properties of the Bernstein density copula estimator for $\alpha$-mixing data. J. Multivariate Anal. 101 1–10.
• [6] Bücher, A. and Volgushev, S. (2013). Empirical and sequential empirical copula processes under serial dependence. J. Multivariate Anal. 119 61–70.
• [7] Chacón, J.E., Duong, T. and Wand, M.P. (2011). Asymptotics for general multivariate kernel density derivative estimators. Statist. Sinica 21 807–840.
• [8] Charpentier, A., Fermanian, J.-D. and Scaillet, O. (2007). The estimation of copulas: Theory and practice. In Copulas: From Theory to Application in Finance (J. Rank, ed.) 35–60. London: Risk Publications.
• [9] Chen, S.X. (1999). Beta kernel estimators for density functions. Comput. Statist. Data Anal. 31 131–145.
• [10] Chen, S.X. and Huang, T.-M. (2007). Nonparametric estimation of copula functions for dependence modelling. Canad. J. Statist. 35 265–282.
• [11] Chen, X., Fan, Y., Pouzo, D. and Ying, Z. (2010). Estimation and model selection of semiparametric multivariate survival functions under general censorship. J. Econometrics 157 129–142.
• [12] Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés. Un test non paramétrique d’indépendance. Acad. Roy. Belg. Bull. Cl. Sci. (5) 65 274–292.
• [13] Denuit, M., Purcaru, O. and Van Keilegom, I. (2006). Bivariate Archimedean copula modelling for censored data in non-life insurance. J. Actuar. Pract. 13 5–32.
• [14] Duong, T. and Hazelton, M.L. (2003). Plug-in bandwidth matrices for bivariate kernel density estimation. J. Nonparametr. Stat. 15 17–30.
• [15] Duong, T. and Hazelton, M.L. (2005). Convergence rates for unconstrained bandwidth matrix selectors in multivariate kernel density estimation. J. Multivariate Anal. 93 417–433.
• [16] Embrechts, P. (2009). Copulas: A personal view. J. Risk Insur. 76 639–650.
• [17] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability 66. London: Chapman & Hall.
• [18] Fermanian, J.-D. (2005). Goodness-of-fit tests for copulas. J. Multivariate Anal. 95 119–152.
• [19] Fermanian, J.-D., Radulović, D. and Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli 10 847–860.
• [20] Fermanian, J.-D. and Scaillet, O. (2003). Nonparametric estimation of copulas for time series. J. Risk 5 25–54.
• [21] Frees, E.W. and Valdez, E.A. (1998). Understanding relationships using copulas. N. Am. Actuar. J. 2 1–25.
• [22] Geenens, G. (2014). Probit transformation for kernel density estimation on the unit interval. J. Amer. Statist. Assoc. 109 346–358.
• [23] Geenens, G., Charpentier, A. and Paindaveine, D. (2016). Supplement to “Probit transformation for nonparametric kernel estimation of the copula density.” DOI:10.3150/15-BEJ798SUPP.
• [24] Genest, C., Masiello, E. and Tribouley, K. (2009). Estimating copula densities through wavelets. Insurance Math. Econom. 44 170–181.
• [25] Genest, C. and Rémillard, B. (2004). Tests of independence and randomness based on the empirical copula process. TEST 13 335–370.
• [26] Genest, C., Rémillard, B. and Beaudoin, D. (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance Math. Econom. 44 199–213.
• [27] Genest, C. and Segers, J. (2010). On the covariance of the asymptotic empirical copula process. J. Multivariate Anal. 101 1837–1845.
• [28] Gijbels, I. and Mielniczuk, J. (1990). Estimating the density of a copula function. Comm. Statist. Theory Methods 19 445–464.
• [29] Gudendorf, G. and Segers, J. (2012). Nonparametric estimation of multivariate extreme-value copulas. J. Statist. Plann. Inference 142 3073–3085.
• [30] Härdle, W.K. and Okhrin, O. (2010). De copulis non est disputandum. Copulae: An overview. AStA Adv. Stat. Anal. 94 1–31.
• [31] Hjort, N.L. and Jones, M.C. (1996). Locally parametric nonparametric density estimation. Ann. Statist. 24 1619–1647.
• [32] Janssen, P., Swanepoel, J. and Veraverbeke, N. (2014). A note on the asymptotic behavior of the Bernstein estimator of the copula density. J. Multivariate Anal. 124 480–487.
• [33] Joe, H. (1997). Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73. London: Chapman & Hall.
• [34] Jones, M.C. and Signorini, D.F. (1997). A comparison of higher-order bias kernel density estimators. J. Amer. Statist. Assoc. 92 1063–1073.
• [35] Kauermann, G., Schellhase, C. and Ruppert, D. (2013). Flexible copula density estimation with penalized hierarchical B-splines. Scand. J. Stat. 40 685–705.
• [36] Klugman, S.A. and Parsa, R. (1999). Fitting bivariate loss distributions with copulas. Insurance Math. Econom. 24 139–148.
• [37] Li, B. and Genton, M.G. (2013). Nonparametric identification of copula structures. J. Amer. Statist. Assoc. 108 666–675.
• [38] Loader, C. (1999). Local Regression and Likelihood. New York: Springer.
• [39] Loader, C.R. (1996). Local likelihood density estimation. Ann. Statist. 24 1602–1618.
• [40] Lopez-Paz, D., Hernández-Lobato, J.M. and Schölkopf, B. (2013). Semi-supervised domain adaptation with non-parametric copulas. Unpublished manuscript.
• [41] Mack, Y.P. and Rosenblatt, M. (1979). Multivariate $k$-nearest neighbor density estimates. J. Multivariate Anal. 9 1–15.
• [42] Nelsen, R.B. (2006). An Introduction to Copulas, 2nd ed. New York: Springer.
• [43] 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.
• [44] Qu, L. and Yin, W. (2012). Copula density estimation by total variation penalized likelihood with linear equality constraints. Comput. Statist. Data Anal. 56 384–398.
• [45] Salmon, F. (2009). Recipe for disaster: The formula that killed wall street. Wired Magazine, February 23.
• [46] Scaillet, O. (2007). Kernel-based goodness-of-fit tests for copulas with fixed smoothing parameters. J. Multivariate Anal. 98 533–543.
• [47] Segers, J. (2012). Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli 18 764–782.
• [48] Shen, X., Zhu, Y. and Song, L. (2008). Linear B-spline copulas with applications to nonparametric estimation of copulas. Comput. Statist. Data Anal. 52 3806–3819.
• [49] Sklar, M. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 229–231.
• [50] Tsukahara, H. (2005). Semiparametric estimation in copula models. Canad. J. Statist. 33 357–375.
• [51] Wand, M.P. and Jones, M.C. (1995). Kernel Smoothing. Monographs on Statistics and Applied Probability 60. London: Chapman & Hall.

#### Supplemental materials

• Supplement to “Probit transformation for nonparametric kernel estimation of the copula density”. An appendix consisting of the proofs of the different theoretical results is available as supplementary material.