Special Issue 2022 – Call for Papers
A New Role for Statistics: The Joint Special Issue of "Statistics in Transition New Series" (SiTns) and "Statystyka Ukraïny" (SU)
Abdelmalek Gagui https://orcid.org/0000- 0002-2715-4304 , Abdelhak Chouaf

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This paper deals with the conditional hazard estimator of a real response where the variable is given a functional random variable (i.e it takes values in an infinite-dimensional space). Specifically, we focus on the functional index model. This approach offers a good compromise between nonparametric and parametric models. The principle aim is to prove the asymptotic normality of the proposed estimator under general conditions and in cases where the variables satisfy the strong mixing dependency. This was achieved by means of the kernel estimator method, based on a single-index structure. Finally, a simulation of our methodology shows that it is efficient for large sample sizes.


single functional index, conditional hazard function, nonparametric estimation, ?-mixing dependency, asymptotic normality, functional data


Ait Saidi, A., Ferraty, F., Kassa, P. and Vieu, P., (2005). Single functional index model for a time series. Rev. Roumaine Math. Pures Appl. 50, pp. 321–330.

Ait Saidi, A., Ferraty, F., Kassa, P. and Vieu, P., (2008). Cross-validated estimations in the single functional index model. Statistics. Vol. 42, No. 6, pp. 475–494.

Ait Saidi, A. and Mecheri, K., (2016). The conditional cumulative distribution function in single functional index model. Comm. Statist. Theory Methods. 45, pp. 4896–4911.

Arfi, M., (2013). Nonparametric Estimation for the Hazard Function. Communications in Statistics - Theory and Methods 42, pp. 2543–2550.

Attaoui, S., (2014). Strong uniform consistency rates and asymptotic normality of conditional density estimator in the single functional index modeling for time series data.

AStA - Advances in Statistical Analysis 98, pp. 257–286.

Attaoui, S., laksaci, A. and Ould Said, F., (2011). A note on the conditional density estimate in the single functional index model. Statist. Probab. Lett., 81, pp. 45–53.

Bagkavos, D., (2011). Local linear hazard rate estimation and bandwidth selection . Annals of the Institute of Statistical Mathematics 63, pp. 1019–1046.

Bouraine, M., Ait Saidi, A., Ferraty, F. and Vieu, P., (2010). Choix optimal de l’indice Multi-fonctionnel: Methode de validation crois´ee. Rev. Roumaine Math.Pures Appl.. 55, pp. 355–367.

Burba, F., Ferraty, F. and Vieu, P., (1996). Convergence of k nearest neighbor kernel estimator in nonparametric functional regression. Comptes Rendus Mathematique. 346, pp. 339–342.

Collomb, G., Hassani, S., Sarda, P. and Vieu, P., (1985). Estimation non parametrique de la fonction de hasard pour des observations dependentes. Statistique et Analyse des Donn´ees 10, pp. 42–49.

Dabo-Niang, S., Kaid, Z. and Laksaci, A., (2012). On spatial conditional mode estimation for a functional regressor. Statistics and probability letters,. 82, pp. 1413–1421.

Delecroix, M. , Yazourh , O., ( 1992 ). Estimation de la fonction de hazard en pr´esence de censure droite. M´ethode des fonctions orthogonales. Statistique et Analyse des Donn´ees 16, pp. 39–62 .

Estev`ez-P´erez, G., Quintela-del-R, A. and Vieu, P., (2002). Convergence rate for crossvalidatory bandwith in kernel hazard estimation from dependent samples. J. Statist. Plann. Inference 104, pp. 1–30.

Ferraty, F. and Vieu, P., (2006). Nonparametric functional data analysis. Theory and Practice. Springer Series in Statistics. New York.

Ferraty, F., Laksaci , A. and Vieu, P., (2005). Functional times series prediction via conditional mode, C. R., Math. Acad. Sci. Paris 340(5), pp. 389–392.

Ferraty, F., Laksaci , A. and Vieu, P., (2006). Estimating some characteristics of the conditional distribution in nonparametric functional models, Stat. Inf. Stoch. Proc. 9, pp. 47–76.

Ferraty, F., Laksaci, A., Tadj, A., and Vieu, P.,(2010). Rate of uniform consistency for nonparametric estimates with functional variables, Journal of Statistical Planning and Inference. 140, pp. 335–352.

H¨ardle, W., Hall, P. and Ichumira, H., (1993). Optimal smoothing in single-index models Ann. Statist.. 27, pp. 157–178.

Hristache, M., Juditsky, A., and Spokoiny, V., (2001). Direct estimation of the index coefficient in the single-index model. Ann. Statist., 29 , pp. 595–623.

Laksaci, A. and Mechab, B., (2010). Estimation nonparam´etrique de la fonction de hasard avec variable explicative fonctionnelle : cas des donn´ees spatiales. Rev. Roumaine Math. Pures Appl. 55, pp. 35–51.

Laksaci, A. and Maref, F., (2009). Nonparametric estimation of conditional quantiles for functional and spatial dependent variables. Comptes Rendus Mathematique. 347, pp. 1075–1080.

Lecoutre, J. P. and Ould-Said, E.,(1992). Estimation de la densit´e et de la fonction de hasard conditionnelle pour un processus fortement m´elangeant avec censure. C.R. Math. Acad. Sci.Paris. 314, pp. 295–300.

Li, J. and Tran, L. T., (2007). Hazard rate estimation on random fields, J. Multivariate Anal. 98 , pp. 1337–1355.

Masry, E., (2005). Nonparametric regression estimation for dependent functional data: asymptotic normality, Stoch. Process. Appl. 115, pp. 155–177.

Patil, P. N. , Wells M. T. and Marron J. S., (1993). Some heuristics of kernel based estimators of ratio functions. Journal of Nonparametric Statistics 4, pp. 203–209.

Quintela-Del-R´io A., (2008). Hazard function given a functional variable: Non-parametric estimation under strong mixing conditions. Journal of Nonparametric Statistics 5, pp. 413–430.

Roussas, G. G., (1968). On some properties of nonparametric estimates of probability density functions Bull. Soc. Math. Greece (N. S.) 9, pp. 29–43 .

Tabti, H. and Ait Saidi, A., (2017). Estimation and simulation of conditional hazard function in the quasi-associated framework when the observations are linked via a functional single-index structure, Communications in Statistics - Theory and Methods 47, pp. 816–838 .

Tanner, M. A. andWong,W. H., (1983). The Estimation of the Hazard Function from Randomly Censored Data by the Kernel Method The Annals of Statistics 11, pp. 989–993 .

Watson G. S. and Leadbetter M. R., (1964). Hazard Analysis. I Biometrika 51, pp. 175– 184 .

Youndj´e, ´ E., Sarda, P. and Vieu, P., (1996). Optimal smooth hazard estimates. Test. 5, pp. 379–394.

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