Nonparametric density and survival function estimation in the multiplicative censoring model

Consider the multiplicative censoring model given by $$Y_i=X_iU_i$$ Y i = X i U i , $$i=1, \ldots ,n$$ i = 1 , … , n where $$(X_i)$$ ( X i ) are i.i.d. with unknown density f on $${\mathbb {R}}$$ R , $$(U_i)$$ ( U i ) are i.i.d. with uniform distribution $${\mathcal {U}}([0,1])$$ U ( [ 0 , 1 ] ) and $$(U_i)$$ ( U i ) and $$(X_i)$$ ( X i ) are independent sequences. Only the sample $$(Y_i)$$ ( Y i ) is observed. We study nonparametric estimators of both the density f and the corresponding survival function $$\bar{F}$$ 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 $$X_i$$ X i s are nonnegative, using kernels fitted for $${\mathbb {R}}^+$$ 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.