Nonparametric density and survival function estimation in the multiplicative censoring model

Consider the multiplicative censoring model given by , where are i.i.d. with unknown density f on , are i.i.d. with uniform distribution and and are independent sequences. Only the sample is observed. We study nonparametric estimators of both the density f and the corresponding survival function . 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 s are nonnegative, using kernels fitted for -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.