## Electronic Journal of Statistics

### Multivariate and functional covariates and conditional copulas

#### Abstract

In this paper the interest is to estimate the dependence between two variables conditionally upon a covariate, through copula modelling. In recent literature nonparametric estimators for conditional copula functions in case of a univariate covariate have been proposed. The aim of this paper is to nonparametrically estimate a conditional copula when the covariate takes on values in more complex spaces. We consider multivariate covariates and functional covariates. We establish weak convergence, and bias and variance properties of the proposed nonparametric estimators. We also briefly discuss nonparametric estimation of conditional association measures such as a conditional Kendall’s tau. The case of functional covariates is of particular interest and challenge, both from theoretical as well as practical point of view. For this setting we provide an illustration with a real data example in which the covariates are spectral curves. A simulation study investigating the finite-sample performances of the discussed estimators is provided.

#### Article information

Source
Electron. J. Statist., Volume 6 (2012), 1273-1306.

Dates
First available in Project Euclid: 26 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1343310298

Digital Object Identifier
doi:10.1214/12-EJS712

Mathematical Reviews number (MathSciNet)
MR2988448

Zentralblatt MATH identifier
1295.62031

#### Citation

Gijbels, Irène; Omelka, Marek; Veraverbeke, Noël. Multivariate and functional covariates and conditional copulas. Electron. J. Statist. 6 (2012), 1273--1306. doi:10.1214/12-EJS712. https://projecteuclid.org/euclid.ejs/1343310298

#### References

• [1] Abegaz, F., Gijbels, I. and Veraverbeke, N. (2012). Semiparametric estimation of conditional copulas., J. Multivariate Anal. 110 43–73.
• [2] Acar, E. F., Craiu, R. V. and Yao, F. (2011). Dependence calibration in conditional copulas: a nonparametric approach., Biometrics 67 445–453.
• [3] Baíllo, A. and Grané, A. (2009). Local linear regression for functional predictor and scalar response., J. Multivariate Anal. 100 102–111.
• [4] Barrientos-Marin, J., Ferraty, F. and Vieu, P. (2010). Locally modelled regression and functional data., J. Nonparametr. Stat. 22 617–632.
• [5] Berlinet, A., Elamine, A. and Mas, A. (2011). Local linear regression for functional data., Ann. Inst. Statist. Math. 63 1047–1075.
• [6] Borggaard, C. and Thodberg, H. H. (1992). Optimal minimal neural interpretation of spectra., Analytical Chemistry 64 545–551.
• [7] Burba, F., Ferraty, F. and Vieu, P. (2009). $k$-Nearest Neighbour method in functional nonparametric regression., J. Nonparametr. Stat. 21 453–469.
• [8] Ferraty, F., Van Keilegom, I. and Vieu, P. (2010). On the Validity of the Bootstrap in Non-Parametric Functional Regression., Scand. J. Statist. 37 286–306.
• [9] Ferraty, F. and Vieu, P. (2002). The functional nonparametric model and application to spectrometric data., Computation. Stat. 17 545–564.
• [10] Ferraty, F., Mas, A. and Vieu, P. (2007). Nonparametric regression on functional data: inference and practical aspects., Aust. N. Z. J. Stat. 49 267–286.
• [11] Ferraty, F. and Vieu, P. (2006)., Nonparametric functional data analysis. Theory and practice. Springer, New York.
• [12] Gijbels, I., Veraverbeke, N. and Omelka, M. (2011). Conditional copulas, association measures and their application., Comput. Stat. Data An. 55 1919–1932.
• [13] Hafner, C. M. and Reznikova, O. (2010). Efficient estimation of a semiparametric dynamic copula model., Comput. Stat. Data An. 54 2609–2627.
• [14] Hall, P., Müller, H. G. and Yao, F. (2009). Estimation of functional derivatives., Ann. Statist. 37 3307–3329.
• [15] Nelsen, R. B. (2006)., An Introduction to Copulas. Springer, New York Second Edition.
• [16] Ruppert, D. and Wand, M. P. (1994). Multivariate locally weighted least squares regression., Ann. Statist. 22 1346–1370.
• [17] Segers, J. (2012). Weak convergence of empirical copula processes under nonrestrictive smoothness assumptions., Bernoulli 18 764–782.
• [18] van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes. Springer, New York.
• [19] Veraverbeke, N., Omelka, M. and Gijbels, I. (2011). Estimation of a conditional copula and association measures., Scand. J. Statist. 38 766–780.