Article
Full entry |
PDF
(1.7 MB)
Feedback
PDF
(1.7 MB)
Feedback
Keywords:
twice censoring; associated data; kernel estimators; density; failure rate; asymptotic normality
twice censoring; associated data; kernel estimators; density; failure rate; asymptotic normality
Summary:
In this work, we study kernel estimators of the density and the failure rate in the context of association and twice censoring. Association is a very interesting type of dependence given the fact that it covers many useful random processes. In this context, we assume that we have at disposal a sample of strictly stationary and associated copies of the variable of interest. Furthermore, this variable is right censored by another variable, the minimum of the two variables is left censored and the three latent variables are independent. Under these assumptions, we establish the asymptotic normality of kernel estimators of the density and the failure rate. Then, we illustrate the performance and the Gaussian behavior of these estimators through a simulation study. The results of this study show the good quality of estimation and the good behavior of the studied asymptotic distributions with respect to the standard normal distribution. We also show the performance of our estimators using an application on a real data set.
References:
[1] Bagai, I., Rao, B. L. S. Prakasa: Estimation of the survival function for stationary associated processes. Statist. Probab. Lett. 12 (1991), 385-391. DOI
[2] Bagai, I., Rao, B. L. S. Prakasa: Kernel-type density and failure rate estimation for associated sequences. Ann. Inst. Statist. Math. 47 (1995), 253-266. DOI
[3] Billingsley, P.: Probability and Measure. Wiley Series in Probability and Mathematical Statistics, John Wiley and Sons Inc., New York 1995.
[4] Boukeloua, M.: Rates of mean square convergence of density and failure rate estimators under twice censoring. Statist. Probab. Lett. 106 (2015), 121-128. DOI
[5] Boukeloua, M.: Testing local hypotheses with different types of incomplete data. TWMS Journal of Applied and Engineering Mathematics 14 (2024), 1374-1389.
[6] Boukeloua, M., Messaci, F.: Asymptotic normality of kernel estimators based upon incomplete data. J. Nonparametr. Statist. 28 (2016), 469-486. DOI
[7] Boukeloua, M., Toumi, S., Idiou, N., Benatia, F.: Kernel estimation of the copula function under twice censoring. Stud. Engrg. Exact Sci. 5 (2024), e12036 DOI
[8] Bulinski, A., Shashkin, A.: Limit Theorems for Associated Random Fields and Relayed Systems. World Scientific, Singapore 2007.
[9] Cai, Z., Roussas, G. G.: Kaplan-Meier estimator under association. J. Multivar. Analysis 67 (1998), 318-348. DOI
[10] Dewan, I., Rao, B. L. S. Prakasa: A general method of density estimation for associated random variables. J. Nonparametr. Statist. 10 (1999), 405-420. https://doi.org/10.1080/10485259908832769 (DOI:10.1080/10485259908832769) DOI 10.1080/10485259908832769
[11] Dewan, I., Rao, B. L. S. Prakasa: Non-uniform and uniform Berry-Esseen type bounds for stationary associated sequences. J. Nonparametr. Statist. 17 (2005), 217-235. DOI
[12] Doukhan, P., Louhichi, S.: A new weak dependence condition and applications to moment inequalities. Stochast. Processes Appl. 84 (1999), 313-342. DOI
[13] Ferrani, Y.: Sur l'estimation non paramétrique de la densité et du mode dans les modèles de données incomplètes et associées. Ph.D. Thesis, Université du Littoral Côte d'Opale France; Université des Sciences et de la Technologie Houari-Boumediene Algeria 2014.
[14] Ferrani, Y., Sa\"id, E. Ould, Tatachak, A.: On kernel density and mode estimates for associated and censored data. Commun. Statist. Theory Methods 45 (2016), 1853-1862. DOI
[15] Julià, O., Gómez, G.: Simultaneous marginal survival estimators when doubly censored data is present. Lifetime Data Analysis 17 (2011), 347-372. DOI
[16] Kitouni, A., Boukeloua, M., Messaci, F.: Rate of strong consistency for nonparametric estimators based on twice censored data. Statist. Probab. Lett. 96 (2015), 255-261. DOI
[17] Marshall, A. W., Olkin, I.: A multivariate exponential distribution. J. Amer. Statist. Assoc. 62 (1967), 30-44. DOI | Zbl 0147.38106
[18] Patilea, V., Rolin, J. M.: Product-limit estimators of the survival function with twice censored data. Ann. Statist. 34 (2006), 925-938. DOI
[19] Pitt, L. D.: Positively correlated normal variables are associated. Ann. Probab. 10 (1982), 496-499. DOI
[20] Roussas, G. G.: Kernel estimates under association: Strong uniform consistency. Statist. Prob. Lett. 12 (1991), 393-403. DOI
[21] Roussas, G. G.: Asymptotic normality of the kernel estimate of a probability density function under association. Statist. Probab. Lett. 50 (2000), 1-12. DOI
[22] Saihi, F., Boukeloua, M., Leulmi, S.: Non parametric estimation in the presence of associated and twice censored data. Commun.Statist. Theory Methods 54 (2025), 6536-6561. DOI
[23] Toumi, S., Boukeloua, M., Idiou, N., Benatia, F.: Nonparametric estimation of the copula function with bivariate twice censored data. Boletim da Sociedade Paranaense de Matemática 42 (2024), 1-22. DOI
[24] Yu, H.: A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated sequences. Probab. Theory Related Fields 95 (1993), 357-370. DOI
Search
Partner of
