## Electronic Journal of Statistics

### Semiparametric copula quantile regression for complete or censored data

#### Abstract

When facing multivariate covariates, general semiparametric regression techniques come at hand to propose flexible models that are unexposed to the curse of dimensionality. In this work a semiparametric copula-based estimator for conditional quantiles is investigated for both complete or right-censored data. In spirit, the methodology is extending the recent work of Noh, El Ghouch and Bouezmarni [34] and Noh, El Ghouch and Van Keilegom [35], as the main idea consists in appropriately defining the quantile regression in terms of a multivariate copula and marginal distributions. Prior estimation of the latter and simple plug-in lead to an easily implementable estimator expressed, for both contexts with or without censoring, as a weighted quantile of the observed response variable. In addition, and contrary to the initial suggestion in the literature, a semiparametric estimation scheme for the multivariate copula density is studied, motivated by the possible shortcomings of a purely parametric approach and driven by the regression context. The resulting quantile regression estimator has the valuable property of being automatically monotonic across quantile levels. Additionally, the copula-based approach allows the analyst to spontaneously take account of common regression concerns such as interactions between covariates or possible transformations of the latter. From a theoretical prospect, asymptotic normality for both complete and censored data is obtained under classical regularity conditions. Finally, numerical examples as well as a real data application are used to illustrate the validity and finite sample performance of the proposed procedure.

#### Article information

Source
Electron. J. Statist., Volume 11, Number 1 (2017), 1660-1698.

Dates
First available in Project Euclid: 25 April 2017

https://projecteuclid.org/euclid.ejs/1493107294

Digital Object Identifier
doi:10.1214/17-EJS1273

Mathematical Reviews number (MathSciNet)
MR3639560

Zentralblatt MATH identifier
06715787

#### Citation

De Backer, Mickaël; El Ghouch, Anouar; Van Keilegom, Ingrid. Semiparametric copula quantile regression for complete or censored data. Electron. J. Statist. 11 (2017), no. 1, 1660--1698. doi:10.1214/17-EJS1273. https://projecteuclid.org/euclid.ejs/1493107294

#### References

• [1] Brechmann, E. C. (2010). Truncated and Simplified Regular Vines and Their Applications Masterarbeit, Technische Universität, München.
• [2] Bücher, A., El Ghouch, A. and Van Keilegom, I. (2014). Single-index Quantile Regression models for Censored Data., Submitted.
• [3] Charpentier, A., Fermanian, J. D. and Scaillet, O. (2006)., The estimation of copulas: Theory and practice. J. Rank, ed., Risk Books.
• [4] Chaudhuri, P. (1991). Global nonparametric estimation of conditional quantile functions and their derivatives., Journal of Multivariate Analysis 39 246–269.
• [5] Chen, S. X. and Huang, T. M. (2007). Nonparametric estimation of copula functions for dependence modelling., Canad. J. Statist. 35 265–282.
• [6] Chen, K. and Lo, S. H. (1997). On the rate of uniform convergence of the product-limit estimator: strong and weak laws., The Annals of Statistics 25 1050–1087.
• [7] Czado, C. (2010)., Workshop on Copula Theory and Its Applications. Springer.
• [8] Dette, H., Van Hecke, R. and Volgushev, S. (2014). Some comments on copula-based regression., J. Amer. Statist. Assoc. 109(507) 1319–1324.
• [9] Dißmann, J., Brechmann, E. C., Czado, C. and Kurowicka, D. (2013). Selecting and Estimating Regular Vine Copulae and Application to Financial Returns., Comput. Stat. Data Anal. 59(0) 52–69.
• [10] El Ghouch, A. and Van Keilegom, I. (2009). Local Linear quantile Regression with Dependent Censored Data., Statistica Sinica 19 1621–1640.
• [11] Elsner, J. B., Kossin, J. P. and Jagger, T. H. (2008). The increasing intensity of the strongest tropical cyclones., Nature 455(7209) 92–95.
• [12] Embrechts, P. (2009). Copulas: A Personal View., Journal of Risk and Insurance 76(3) 639–650.
• [13] Feng, X., He, X. and Hu, J. (2011). Wild bootstrap for quantile regression., Biometrika 98 995–999.
• [14] Geenens, G., Charpentier, A. and Paindaveine, D. (2014). Probit transformation for nonparametric kernel estimation of the copula density., arXiv:1404.4414 [stat.ME].
• [15] Genest, C. and Nešlehová, J. (2007). A Primer on Copulas for Count Data., The ASTIN Bulletin 37(2) 475–515.
• [16] Gijbels, I. and Mielniczuk, J. (1990). Estimating the density of a copula function., Communications in Statistics – Theory and Methods 19(2) 445–464.
• [17] Hjort, N. L. and Pollard, D. (1993). Asymptotics for minimisers of convex processes Technical Report, Yale, University.
• [18] Hobæk Haff, I., Aas, K. and Frigessi, A. (2010). On the simplified pair-copula construction – simply useful or too simplistic?, Journal of Multivariate Analysis 101(5) 1296–1310.
• [19] Hobæk Haff, I. and Segers, J. (2015). Nonparametric estimation of pair-copula constructions with the empirical pair-copula., Computational Statistics & Data Analysis 84 1–13.
• [20] Hofert, M. and Pham, D. (2013). Densities of nested Archimedian copulas., Journal of Multivariate Analysis 118 37–52.
• [21] Joe, H. (2014)., Dependence Modeling with Copulas. Chapman & Hall/CRC Monographs on Statistics & Applied Probability.
• [22] Koenker, R. (2005)., Quantile Regression. Cambridge Univ. Press.
• [23] Koenker, R. and Basset, G. J. (1978). Regression quantiles., Econometrica 46(1) 33–50.
• [24] Koenker, R. and Bilias, Y. (2001). Quantile regression for duration data: A reappraisal of the Pensylvania employment bonus experiments., Empirical Economics 26 199–220.
• [25] Koenker, R. and Geling, O. (2001). Reappraising medfly longevity: a quantile regression survival analysis., J. Amer. Statist. Assoc. 96 458–468.
• [26] Langholz, B. and Goldstein, L. (1996). Risk set sampling in epidemiologic cohort studies., Statistical Science 35–53.
• [27] Leng, C. and Tong, X. (2013). A quantile regression estimator for censored data., Bernoulli 19(1) 344–361.
• [28] Li, Q., Lin, J. and Racine, J. S. (2013). Optimal bandwidth selection for nonparametric conditional distribution and quantile functions., Journal of Business & Economic Statistics 31 57–65.
• [29] Lubin, J. H., Boice, J. D., Edling, C., Hornung, R. W., Howe, G. R., Kunz, E., Kusiak, R. A., Morrison, H. I., Radford, E. P., Samet, J. M. et al. (1995). Lung cancer in radon-exposed miners and estimation of risk from indoor exposure., Journal of the National Cancer Institute 87 817–827.
• [30] Martins-Filho, C. and Yao, F. (2006). A Note on the use of V and U statistics in nonparametric models of regression., Annals of the Institute of Statistical Mathematics 58 389–406.
• [31] Nagler, T. (2014). Kernel Methods for Vine Copula Estimation Masterarbeit, Technische Universität, München.
• [32] Nagler, T. and Czado, C. (2016). Evading the curse of dimensionality in multivariate kernel density estimation with simplified vines., Journal of Multivariate Analysis 151 69–89.
• [33] Nelsen, R. (2006)., An Introduction to Copulas. Springer, New York.
• [34] Noh, H., El Ghouch, A. and Bouezmarni, T. (2013). Copula-Based Regression Estimation and Inference., J. Amer. Statist. Assoc. 108 676–688.
• [35] Noh, H., El Ghouch, A. and Van Keilegom, I. (2015). Semiparametric Conditional Quantile Estimation through Copula-Based Multivariate Models., Journal of Business and Economic Statistics 33(2) 167–178.
• [36] Oh, D. H. and Patton, A. (2012). Modelling dependence in high dimensions with factor copulas., Manuscript, Duke University.
• [37] Portnoy, S. (2003). Censored Regression Quantiles., J. Amer. Statist. Assoc. 98(464) 1001–1012.
• [38] Portnoy, S. and Koenker, R. (1997). The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators., Statistical Science 12 279–300.
• [39] Powell, J. L. (1986). Censored Regression Quantiles., J. Econometrics. 32 143–155.
• [40] Serfling, R. J. (1980)., Approximation Theorems of Mathematical Statistics. Wiley Series in Probability and Statistics – Applied Probability and Statistics Section Series. Wiley.
• [41] Sklar, M. (1959). Fonctions de répartition à $n$ dimensions et leurs marges., Publ. Inst. Stitst. Univ. Paris 8 229–231.
• [42] Spokoiny, V., Wang, W. and Härdle, W. K. (2013). Local quantile regression., Journal of Statistical Planning and Inference 143 1109–1129.
• [43] Stöber, J., Joe, H. and Czado, C. (2013). Simplified pair copula constructions – limitations and extensions., Journal of Multivariate Analysis 119(0) 101–118.
• [44] R Core Team (2014). R: A Language and Environment for Statistical Computing R Foundation for Statistical Computing, Vienna, Austria.
• [45] Wang, H. J. and Wang, L. (2009). Locally weighted censored quantile regression., J. Amer. Statist. Assoc. 104 1117–1128.
• [46] Wu, T., Yu, K. and Yu, Y. (2010). Single-index quantile regression., Journal of Multivariate Analysis 101 1607–1621.
• [47] Ying, Z., Jung, S. H. and Wei, L. J. (1995). Survival analysis with median regression models., J. Amer. Statist. Assoc. 90 178–184.
• [48] Zhu, L., Huang, M. and Li, R. (2012). Semiparametric quantile regression with high-dimensional covariates., Statistica Sinica 22 1379–1401.