Nonparametric density and survival function estimation in the multiplicative censoring model

Abstract

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.