Electronic Journal of Statistics

Nonparametric conditional density estimation for censored data based on a recursive kernel

Salah Khardani and Sihem Semmar

Full-text: Open access


Consider a regression model in which the response is subject to random right censoring. The main goal of this paper concerns the kernel estimation of the conditional density function in the case of censored interest variable. We employ a recursive version of the Nadaraya-Watson estimator in this context. The uniform strong consistency of the recursive kernel conditional density estimator is derived. Also, we prove the asymptotic normality of this estimator.

Article information

Electron. J. Statist., Volume 8, Number 2 (2014), 2541-2556.

First available in Project Euclid: 9 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G07: Density estimation 62G08: Nonparametric regression 62G20: Asymptotic properties 62H12: Estimation

Asymptotic normality censored data conditional density kernel estimator recursive estimation Kaplan–Meier estimator uniform almost sure convergence


Khardani, Salah; Semmar, Sihem. Nonparametric conditional density estimation for censored data based on a recursive kernel. Electron. J. Statist. 8 (2014), no. 2, 2541--2556. doi:10.1214/14-EJS960. https://projecteuclid.org/euclid.ejs/1418134263

Export citation


  • [1] Ahmad, I., Lin, P.E. (1976). Nonparametric sequential estimation of a multiple regression function, Bull. Math. Statist., 17, 63–75.
  • [2] Amiri, A. (2009). Sur une famille paramétrique d’estimateurs séquentiels de la densité pour un processus fortemement mélangeant, C. R. Acad. Sci. Paris, Ser I, 347, 309–314.
  • [3] Amiri, A. (2012). Recursive regression estimators with application to nonparametric prediction, J. Nonparam. Statist., 24(1), 169–186.
  • [4] Amiri, A. (2013). Asymptotic normality of recursive estimators under strong mixing conditions, arXiv:1211.5767v2.
  • [5] Beran, R. (1981). Nonparametric regression with randomly censored survival data, Technical university of California, Berkeley.
  • [6] Carroll, J. (1976). On sequential density estimation, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36, 137–151.
  • [7] Carbonez, A., Györfi, L., Vander Meulin, E.C. (1995). Partitioning estimates of a regression function under random censoring, Statist. & Decisions, 13, 21–37.
  • [8] Dabrowska, D.M. (1987). Nonparametric regression with censored survival time data, Scandi. J. Statist., 14, 181–197.
  • [9] Dabrowska, D.M. (1989). Uniform consistency of the kernel conditional Kaplan Meier estimate, Ann. Statist, 17, 1157–1167.
  • [10] Davies, I. (1973). Strong consistency of a sequential estimator of a probability density function, Bull. Math. Statist., 15, 49–54.
  • [11] Deheuvels, P., Einmahl, J.H.J. (2000). Functional limit laws for the increments of Kaplan-Meier product-limit processes and applications, Ann. Probab., 28, 1301–1335.
  • [12] Devroye, L., Wagner, T.J. (1980). On the $L^1$ convergence of kernel estimators of regression functions with application in discrimination, Z. Wahrschein. Verw. Get., 51, 15–25.
  • [13] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58, 13–30.
  • [14] Kaplan, E.L., Meier, P. (1958). Nonparametric estimation from incomplete observations, J. Amer. Statist. Assoc., 53, 457–481.
  • [15] Khardani, S., Lemdani, M., Ould Saïd, E. (2010). Some asymptotic properties for a smooth kernel estimator of the conditional mode under random censorship, J. of the Korean Statistical Society, 39, 455–469.
  • [16] Khardani, S., Lemdani, M., Ould Saïd, E. (2011). Uniform rate of strong consistency for a smooth kernel estimator of the conditional mode for censored time series, J. Stat. Plann. Inference, 141, 3426–3436.
  • [17] Kohler, M., Máthé, K., Pinter, M. (2002). Prediction from randomly Right Censored Data, J. Multivariate Anal., 80, 73–100.
  • [18] Krzÿzak, A. (1992). Global convergence of the recursive kernel regression estimates with applications in classification and nonlinear system estimation, IEEE Trans. Inform. Theory, 38, 1323–1338.
  • [19] Loève, M. (1963)., Probability Theory, New York: Springer-Verlag.
  • [20] Masry, E., Fan, J. (1997). Local polynomial estimation of recursive function for mixing processes, Scandinave Journal of Statistics, 24, 165–179.
  • [21] Nguyen, T., Saracco, J. (2010). Estimation récursive en régression inverse par tranches, Journal de la société française de statistique, 151(2), 19–46.
  • [22] Roussas, G.G. (1990). Nonparametric regression estimation under mixing conditions, Stochastic Process. Appl., 36(1), 107–116.
  • [23] Roussas, G.G., Tran, L.T. (1992). Asymptotic normality of the recursive kernel regression estimate under dependence conditions, Annals of Statist., 20(1), 98–120.
  • [24] Walk, H. (2001). Strong universal pointwise consistency of recursive regression estimates, Ann. Inst. Statist. Math., 53(4), 691–707.
  • [25] Wegman, J., Davies, I. (1979). Remarks on some recursive estimators of a probability density, Ann. Statist., 7, 316–327.
  • [26] Wertz, W. (1985). Sequential and recursive estimators of the probability density, Statistics, 16, 277–295.
  • [27] Wolverton, C., Wagner, T.J. (1969). Asymptotically optimal discriminant functions for pattern classifcation, IEEE Trans. Inform. Theory, 15, 258–265.
  • [28] Yamato, H. (1972). Sequential estimation of a continuous probability density function and mode, Bull. Math. Statist. 14, 1–12.