Consistency rates and asymptotic normality of the high risk conditional for functional data

Abbes Rabhi; Latifa Keddani; Yassine Hammou

doi:10.1515/ausm-2016-0008

.blurhash-client-img { display: none !important; }

Consistency rates and asymptotic normality of the high risk conditional for functional data

Acta Universitatis Sapientiae, Mathematica

Volume 8 (2016): Issue 1 (August 2016)

By: Abbes Rabhi, Latifa Keddani and Yassine Hammou

Open Access

|Aug 2016

Abstract

The maximum of the conditional hazard function is a parameter of great importance in seismicity studies, because it constitutes the maximum risk of occurrence of an earthquake in a given interval of time. Using the kernel nonparametric estimates of the first derivative of the conditional hazard function, we establish uniform convergence properties and asymptotic normality of an estimate of the maximum in the context of independence data.

References

[1] M. Lothaire, Combinatorics on words, Addison-Wesley, Reading, MA, 1983.
Search in Google Scholar
[2] I. A. Ahmad, Uniform strong convergence of the generalized failure rate estimate, Bull. Math. Statist., 17 (1976), 77–84.10.5109/13105
Open DOI Search in Google Scholar
[3] N. Belkhir, A. Rabhi, S. Soltani, Exact Asymptotic Errors of the Hazard Conditional Rate Kernel, J. Stat. Appl. Pro. Lett., 2 (3) (2015), 191–204.
Search in Google Scholar
[4] K. Benhenni, F. Ferraty, M. Rachdi, P. Vieu, Local smoothing regression with functional data, Comput. Statist., 22 (2007), 353–369.10.1007/s00180-007-0045-0
Open DOI Search in Google Scholar
[5] P. Besse, H. Cardot, D. Stephenson, Autoregressive forecasting of some functional climatic variations, Scand. J. Statist., 27 (2000), 673–687.10.1111/1467-9469.00215
Open DOI Search in Google Scholar
[6] P. Besse, J. O. Ramsay, Principal component analysis of sampled curves, Psychometrika, 51 (1986), 285–311.10.1007/BF02293986
Open DOI Search in Google Scholar
[7] D. Bosq, J. P. Lecoutre, Théorie de l’estimation fonctionnelle, ECONOMICA (eds), Paris, 1987.
Search in Google Scholar
[8] H. Cardot, F. Ferraty, P. Sarda, Functional linear model, Statist. Probab. Lett., 45 (1999), 11–22.10.1016/S0167-7152(99)00036-X
Open DOI Search in Google Scholar
[9] J. Damon, S. Guillas, The inclusion of exogenous variables in functional autoregressive ozone forecasting, Environmetrics, 13 (2002), 759–774.10.1002/env.527
Open DOI Search in Google Scholar
[10] P. Doukhan, Mixing: Properties and Examples, Lecture Notes in Statist. 85, Springer-Verlag, NewYork, 1994.10.1007/978-1-4612-2642-0
Open DOI Search in Google Scholar
[11] G. Estvez-Prez, A. Quintela-del-Rio, P. Vieu, Convergence rate for cross-validatory bandwidth in kernel hazard estimation from dependent samples, J. Statist. Plann. Inference., 104 (2002), 1–30.10.1016/S0378-3758(01)00245-2
Open DOI Search in Google Scholar
[12] M. Ezzahrioui, E. Ould-Saïd, Asymptotic normality of a nonparametric estimator of the conditional mode function for functional data, Journal of Nonparametric Statistics., 20(1) (2008), 3–18.10.1080/10485250701541454
Open DOI Search in Google Scholar
[13] M. Ezzahrioui, E. Ould Saïd, Some asymptotic results of a nonparametric conditional mode estimator for functional time series data, Statist. Neerlandica., 64 (2010), 171–201.10.1111/j.1467-9574.2010.00449.x
Open DOI Search in Google Scholar
[14] F. Ferraty, A. Laksaci, P. Vieu, Estimating some characteristics of the conditional distribution in nonparametric functional models. Statist. Inference Stochastic Process, 9 (2006), 47–76.
Search in Google Scholar
[15] F. Ferraty, A. Laksaci, A. Tadj, P. Vieu, Rate of uniform consistency for nonparametric estimates with functional variables, J. Statist. Plann. and Inf., 140 (2010), 335–352.10.1016/j.jspi.2009.07.019
Open DOI Search in Google Scholar
[16] F. Ferraty, A. Mas, P. Vieu, Advances in nonparametric regression for functional variables, Australian and New Zealand Journal of Statistics, 49 (2007), 1–20.10.1111/j.1467-842X.2007.00480.x
Open DOI Search in Google Scholar
[17] F. Ferraty, A. Rabhi, P. Vieu, Estimation non-paramétrique de la fonction de hasard avec variable explicative fonctionnelle, Rev. Roumaine Math. Pures Appl., 53 (2008), 1–18.
Search in Google Scholar
[18] F. Ferraty, P. Vieu, Nonparametric functional data analysis. Springer Series in Statistics, Theory and practice, Springer, New York, 2006.
Search in Google Scholar
[19] T. Gasser, P. Hall, B. Presnell, Nonparametric estimation of the mode of a distribution of random curves, Journal of the Royal Statistical Society, Ser. B., 60 (1998), 681–691.
Search in Google Scholar
[20] J. P. Lecoutre, E. Ould-Saïd, Estimation de la densité et de la fonction de hasard conditionnelle pour un processus fortement mélangeant avec censure, C. R. Math. Acad. Sci. Paris., 314 (1992), 295–300.
Search in Google Scholar
[21] E. Liebscher, Central limit theorem for α-mixing triangular arrays with applications to nonparemetric statistics, Mathematical Methods of Statistics, 10 (2) (2001), 194–214.
Search in Google Scholar
[22] A. Mahiddine, A. A. Bouchentouf, A. Rabhi, Nonparametric estimation of some characteristics of the conditional distribution in single functional index model, Malaya Journal of Matematik (MJM), 2 (4) (2014), 392–410.10.4064/am41-4-2
Open DOI Search in Google Scholar
[23] E. Masry, Non-parametric regression estimation for dependent functional data: Asymptotic normality, Stoch. Process. Appl., 115 (2005), 155–177.10.1016/j.spa.2004.07.006
Open DOI Search in Google Scholar
[24] A. Quintela-del-Rio, Nonparametric estimation of the maximum hazard under dependence conditions, Statist. Probab. Lett., 76 (2006), 1117–1124.10.1016/j.spl.2005.12.009
Open DOI Search in Google Scholar
[25] A. Quintela-del-Rio, Plug-in bandwidth selection in kernel hazard estimation from dependent data, Comput. Stat. Data Anal., 51 (2007), 5800–5812.10.1016/j.csda.2006.10.010
Open DOI Search in Google Scholar
[26] A. Quintela-del-Rio, Hazard function given a functional variable: Nonparametric estimation under strong mixing conditions, J. Nonparametr. Stat., 20 (2008), 413–430.10.1080/10485250802159297
Open DOI Search in Google Scholar
[27] A. Rabhi, S. Soltani, A. Traore, Conditional risk estimate for functional data under strong mixing conditions, Journal of Statistical Theory and Applications, 14 (3) (2015), 301–323.10.2991/jsta.2015.14.3.7
Open DOI Search in Google Scholar
[28] M. Rachdi, P. Vieu, Non-parametric regression for functional data: Automatic smoothing parameter selection, J. Stat. Plan. Inference., 137 (2007), 2784–2801.10.1016/j.jspi.2006.10.001
Open DOI Search in Google Scholar
[29] J. O. Ramsay, B. W. Silverman, Functional Data Analysis, 2nd ed., Springer-Verlag, NewYork, 2005.10.1007/b98888
Open DOI Search in Google Scholar
[30] J. Rice, B. W. Silverman, Estimating the mean and covariance structure non-parametrically when the data are curves, J.R. Stat. Soc. Ser. B., 53 (1991), 233–243.
Search in Google Scholar
[31] E. Rio, Théorie asymptotique des processus aléatoires faiblements dépendant, Mathématiques Application, 22(3) (2000), 331–334.
Search in Google Scholar
[32] M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sci., 42 (1956), 43–47.10.1073/pnas.42.1.43
Open DOI Search in Google Scholar
[33] G. Roussas, Hazard rate estimation under dependence conditions, J. Statist. Plann. Inference., 22 (1989), 81–93.10.1016/0378-3758(89)90067-0
Open DOI Search in Google Scholar
[34] N. D. Singpurwalla, M. Y. Wong, Estimation of the failure rate - A survey of non-parametric methods. Part I: Non-Bayesian methods, Commun. Stat. Theory Math., 12 (1983), 559–588.10.1080/03610928308828479
Open DOI Search in Google Scholar
[35] L. Spierdijk, Non-parametric conditional hazard rate estimation: A local linear approach, Comput. Stat. Data Anal., 52 (2008), 2419–2434.10.1016/j.csda.2007.08.007
Open DOI Search in Google Scholar
[36] G. S. Watson, M. R. Leadbetter, Hazard analysis I, Biometrika, 51 (1964), 175–184.10.1093/biomet/51.1-2.175
Open DOI Search in Google Scholar
[37] G. S. Watson, M. R. Leadbetter, Hazard analysis, Sankhyã, 26 (1964), 101–116.
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.1515/ausm-2016-0008 | Journal eISSN: 2066-7752

Journal RSS Feed

Language: English

Page range: 127 - 149

Submitted on: Nov 11, 2015

Published online: Aug 2, 2016

Published in print: Aug 1, 2016

Published by: Sapientia Hungarian University of Transylvania

In partnership with: Paradigm Publishing Services

Keywords:

almost complete convergence,

asymptotic normality,

conditional hazard function,

functional data,

nonparametric estimation

MSC:

Primary: 62F12,

Secondary: 62G20, 62M09

Related subjects:

Mathematics,

General mathematics

© 2016 Abbes Rabhi, Latifa Keddani, Yassine Hammou, published by Sapientia Hungarian University of Transylvania
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 8 (2016): Issue 1 (August 2016)