Mounir Boumahdi , Idir Ouassou , Mustapha Rachdi

© M. Boumahdi, I. Ouassou, M. Rachdi. Article available under the CC BY-SA 4.0 licence


(English) PDF


This paper presents the estimator of the conditional density function of surrogated scalar response variable given a functional random one. We construct a conditional density function by using the available (true) response data and the surrogate data. Then, we build up some asymptotic properties of the constructed estimator in terms of the almost complete convergences. As a result, we compare our estimator with the classical estimator through the Relatif Mean Square Errors (RMSE). Finally, we end this analysis by displaying the superiority of our estimator in terms of prediction when we are lacking complete data.


Density function, surrogate response, functional variable, almost complete convergence, kernel estimators, scalar response, entropy, semi-metric space


Carroll, R. J., Knickerbocker, R. K., and Wang, C. Y., (1995). Dimension reduction in a semiparametric regression model with errors in covariates. The Annals of Statistics.

Carroll, R. J., Wand, M. P.. (1991). Semiparametric estimation in logistic measurement error models. Journal of the Royal Statistical Society.

Duncan, G. J., Hill, D. H.. (1985). An investigation of the extent and consequences of measurement error in labor-economic survey data. Journal of Labor Economics.

Firas, I., Ali Hajj H., and Rachdi, M., (2019). Regression model for surrogate data in high dimensional statistics. Journal of Communications in Statistics – Theory and Methods.

Ferraty, F., Laksaci, A., Tadj, A., Vieu, P., (2010). Rate of uniform consistency for nonparametric estimates with functional variables. Journal of Statistical Planning and Inference.

Ferraty, F., Vieu, P., (2006). Nonparametric functional data analysis. Theory and practice, NY: Springer Series in Statistics.

Ferraty, F., Vieu, P., (2002). The functional nonparametric model and application to spectrometric data. Comput. Statist.

Ferraty, F. Vieu, P., (2011). Kernel regression estimation for functional data. In The Oxford Handbook of Functional Data Analysis (Ed. F. Ferraty and Y. Romain). Oxford University Press.

Hsing, T., Eubank R., (2015). Theoretical foundations of functional data analysis, with an intro- duction to linear operators. Wiley series in probability and statistics. Chichester, UK: John Wiley and Sons.

Horvath, L., Kokoszka P., (2012). Inference for functional data with applications. New York, NY: Springer Series in Statistics.

Goia, A., Vieu P., (2016). An introduction to recent advances in high/infinite dimensional statistics. Journal of Multivariate Analysis, 146, pp. 1–6.

Kolmogorov A. N., Tikhomirov V. M., (1959). ?-entropy and ?-capacity. Uspekhi Mat. Nauk., 14, pp. 3–86., 2, pp. 277–364.

Lecoutre, J. P., (1990). Uniform consistency of a class of regression function estimators for Banach-space valued random variable. Statist. Probab. Lett.

Loeve, M., (1963). Probability Theory, 3rd ed. Van Nostranr Princeton.

Pepe, M. S., (1992). Inference using surrogate outcome data and validation sample. Biometrika 79.

Rachdi, M., Vieu, P., (2007). Nonparametric regression for functional data: Automatic smoothing parameter selection. Journal of Statistical Planning and Inference.

Wang, Q. H., (2000). Estimation of linear error-in-covariables models with validation data under random censorship. Journal of Multivariate Analysis.

Wang, Q. H., Rao, J. N. K., (2002). Empirical likelihood-based in linear errors in covariables models with validation data. Biometrika.

Wang, Q. H., (2003). Dimension reduction in partly linear error-in-response models with validation data. Journal of Multivariate Analysis.

Wang, Q. H., (2006). Nonparametric regression function estimation with surrogate data and validation sampling. Journal of Multivariate Analysis.

Wang, J. L., Chiou, J. M., and Muller, H. G., (2016). Review of functional data analysis. Annual Review of Statistics and Its Application.

Zhang, J., (2014). Analysis of variance for functional data. Monographs on Statistics and Applied Probability.

Back to top
© 2019–2023 Copyright by Statistics Poland, some rights reserved. Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0) Creative Commons — Attribution-ShareAlike 4.0 International — CC BY-SA 4.0