, Volume 25, Issue 3, pp 570–590 | Cite as

Nonparametric density and survival function estimation in the multiplicative censoring model

  • Elodie Brunel
  • Fabienne Comte
  • Valentine Genon-Catalot
Original Paper


Consider the multiplicative censoring model given by \(Y_i=X_iU_i\), \(i=1, \ldots ,n\) where \((X_i)\) are i.i.d. with unknown density f on \({\mathbb {R}}\), \((U_i)\) are i.i.d. with uniform distribution \({\mathcal {U}}([0,1])\) and \((U_i)\) and \((X_i)\) are independent sequences. Only the sample \((Y_i)\) is observed. We study nonparametric estimators of both the density f and the corresponding survival function \(\bar{F}\). First, kernel estimators are built. Pointwise risk bounds for the quadratic risk are given, and upper and lower bounds for the rates in this setting are provided. Then, in a global setting, a data-driven bandwidth selection procedure is proposed. The resulting estimator has been proved to be adaptive in the sense that its risk automatically realizes the bias-variance compromise. Second, when the \(X_i\)s are nonnegative, using kernels fitted for \({\mathbb {R}}^+\)-supported functions, we propose new estimators of the survival function which are also adaptive. By simulation experiments, we check the good performances of the estimators and compare the two strategies.


Adaptive procedure Bandwidth selection Kernel estimators Multiplicative censoring model 

Mathematics Subject Classification

62G07 62N01 

Supplementary material

11749_2016_479_MOESM1_ESM.pdf (188 kb)
Supplementary material 1 (pdf 187 KB)


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Copyright information

© Sociedad de Estadística e Investigación Operativa 2016

Authors and Affiliations

  • Elodie Brunel
    • 1
  • Fabienne Comte
    • 2
  • Valentine Genon-Catalot
    • 2
  1. 1.Université Montpellier, I3M UMR CNRS 5149MontpellierFrance
  2. 2.Université Paris Descartes, MAP5, UMR CNRS 8145ParisFrance

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